The Triangle and its Properties Worksheet

Multiple Choice Questions

1. Arun’s elder brother Rohan went on a trek with his friends. After reaching a point, they could able to see the mountains. Following the trek, one of the friends clicked the picture which Rohan then showed to Arun. Now, Arun wishes to find the height of the mountain (highlighted in the picture) and draw a particular triangle.
   
   
   Name the type of triangle drawn by Arun to find the height of the mountain.
   (a) Acute angled triangle
   (b) Obtuse angled triangle
   (c) Right angled triangle
   (d) Equilateral triangle
2. During summer, watermelon is the most loved fruit. Shravan’s mother cut the watermelon into triangular pieces and give them to Shravan.
   
   
   Choose any one of the pieces and name the points as A, B and C. Now, if we join these points, then the sides of this triangular piece will be:
   (a) AB and BC only
   (b) AC and BC only
   (c) AB and AC only
   (d) AB, BC and AC
3. Danny owns an agricultural land which looks like a triangle when the corners of the land are connected with straight lines. He desires a seat in the centre where he can observe the entire area. For this, Danny take midpoint (say M) on one of the sides (say AB) of this land and join this midpoint with the corner point not on AB (say C). Then CM will be a/an
   
   
   (a) Altitude of a triangle
   (b) Median of a triangle
   (c) Midpoint of a triangle
   (d) Side of a triangle
4. There are multiple photo frames mounted on a wall.
    
   If the height of the top photo frame with corners points P, Q and R is PM, then PM is the
   (a) Median of triangle PQR
   (b) Altitude of triangle PQR
   (c) Side of triangle PQR
   (d) Bisector of side QR
5. Due to heavy rain, a traffic jam has been occurred in one of the part of a city. The three vehicles – a car, a bus and a motorbike are standing in such a way that they form a triangle (say XYZ).
   Consider the car’s position as XY, bus’s position as YZ and motorbike’s as XZ. When the car move forward, an exterior angle is formed as shown:
   
   
   Then the measure of ∠XZY is _____________.
   (a) 58°
   (b) 68°
   (c) 54°
   (d) 180°
6. The "Mile-a-minute Vine" is one of numerous plants having triangular leaves. It is an annual herbaceous vine called mile-a-minute weed. Its alternating, light green leaves range in size from 4 to 7 cm long to 5 to 9 cm wide, and have an equilateral triangular shape.
   
   What might the measures of the three angles be on any given leaf of this plant?
   (a) 50°, 60° and 70°
   (b) 45°, 45° and 90°
   (c) 30°, 60° and 90°
   (d) 60°, 60° and 60°
7. Neeta has a cavity, therefore her mother advised her to use neem chew sticks rather than a toothbrush. She purchased 12 neem sticks of different sizes.
   
   
   These sticks have been separated into four groups of three sticks, and each group's sticks lengths are as follows:
   Group A: 3 cm, 4 cm, 5 cm
   Group B: 6 cm, 8 cm, 10 cm
   Group C: 8 cm, 15 cm, 23 cm
   Group D: 5 cm, 12 cm, 13 cm
   Which of the following group does not form a right angled triangle?
   (a) Group A
   (b) Group B
   (c) Group C
   (d) Group D
8. There is an emergency staircase on every story of the building. Its foot is 9 feet from the wall, while its peak or top is 12 feet above the floor. What may the staircase's possible length be if there are no twists and it is straight?
   
   
   (a) 3 feet
   (b) Approximately 7 feet
   (c) 14 feet
   (d) 15 feet
9. Aditya, Raj and Sachin are enjoying ice-cream candies while returning from school. After they finished eating, they make a triangle using the sticks left. Aditya and Raj says that the lengths of the sticks they hold are 3 cm and 5 cm. What is the possible length of the stick that Sachin holds?
   
   
   (a) 2 cm
   (b) 4 cm
   (c) 8 cm
   (d) 9 cm
   
   Subjective Questions
   
10. Rajiv used a triangular mould to cut a piece from a clay. He then measures the sides of that triangular piece and found that it is an equilateral triangle. Now keeping the clay stable, he drew one of the medians and then an altitude with the same vertex.
    
    
    State whether the medians of this triangular piece will coincide with its altitudes. Justify your answer.
11. Shweta along with her family went for a picnic to Lonavala. While returning back, she got into a shop where she could find various hand bags. Shweta found a bag which was very attractive because of its triangular shape. She wanted to check if it can fit in her backpack and so took the rough measurements. If the measurements are as follows:
    
    
    Find the measure of ∠PRQ.
12. In order to save space, a triangular table is kept in one of the hall's corners. Now, the two sides of this table are touching the two adjacent walls, and these sides are perpendicular to one another. Find the measure of the two angles that are not in the middle of the two walls if the table's sides that touch the walls are equal.
    
    
13. Rahul entered the school from gate no. 1 and went to his class following a straight line. After the class was over, he first went to canteen and then took an exit from gate no. 1. The path took by him is shown below:
    
    
    Find the value of x and then the measures of the three angles.
14. A worm fell into a paint box and Tarun assists the worm to get out of the box. The worm then begin to crawl on the floor, forming an irregular shape (shown below). Now, if the corner points are connected by a straight line, a triangle is formed with the following measurements as seen below:
    
    
    Based on the data given in the above figure, find the value of x.
15. Gauri’s brother is in 4th standard and so he uses crayons for coloring the drawings made in the drawing book. There are just three crayons of different colors and sizes left and the others have been exhausted. The lengths of these crayons are 2 cm, 3 cm and 5cm.
    
    
    Now, if Gauri tries to create a triangle with these crayons, will she succeed? State the reason.
16. Three children Arjun, Rashmi and Suhani, are playing at the different corners of a park. Arjun and Rashmi are separated by 4.5 cm, Rashmi and Suhani by 5.8 cm, while Arjun and Suhani are separated by 10.2 cm. If the positions of these children are considered to be A, R and S respectively, state whether the triangle ARS may be created. Justify your answer.
    
    
17. Harsh is trying to make a sailing boat using a few sticks not necessarily of different lengths.
    
    
    He takes two sticks that are 5 cm and 12 cm long and arranges them so that they are perpendicular to one other. If the third stick he chooses has its length equal to the length of one of the two sticks picked earlier, will he be able to create a sail for the sailing board?
18. There are multiple triangle-shaped chips in a packet, all with different side lengths. John had gone through more over half of the chips, leaving only four in the packet.
    
    
    He then basically stated that only two of the four chips form a right-angled triangle from observing the chips. These four chips' side lengths are listed below:
    3 cm, 4 cm, 5 cm
    1.2 cm, 3.5 cm, 3.7 cm
    1.6 cm, 6.3 cm, 6.5 cm
    5 cm, 7 cm, 13 cm
    Is John correct? Justify your answer.
19. The two flaps of a roof are designed to be perpendicular to one another. The flaps have lengths of 9 and 12 metres. Is it possible to join these flaps together at their ends with a thread that is 14 metres long? If not, suggest a possible length.
20. Ajay went for camping along with his parents. They attempt to construct a tent out of bamboos of various lengths. To do this, first make a right-angled triangle using three bamboos, and then make another right-angled triangle using the same vertical bamboo of previous triangle. The similar shape must be created on the opposite side of these triangles.
    
    
    If the lengths of the first three bamboos are 8 cm, 15 cm and 18 cm, will they be successful in the first step?

Multiple Choice Questions

1. Arun’s elder brother Rohan went on a trek with his friends. After reaching a point, they could able to see the mountains. Following the trek, one of the friends clicked the picture which Rohan then showed to Arun. Now, Arun wishes to find the height of the mountain (highlighted in the picture) and draw a particular triangle.
   
   
   Name the type of triangle drawn by Arun to find the height of the mountain.
   (a) Acute angled triangle
   (b) Obtuse angled triangle
   (c) Right angled triangle
   (d) Equilateral triangle
   Correct Option: (c) Right angled triangle
   Solution:
   The triangle shown in the picture has one right angle. So, it is a right angled triangle.
2. During summer, watermelon is the most loved fruit. Shravan’s mother cut the watermelon into triangular pieces and give them to Shravan.
   
   
   Choose any one of the pieces and name the points as A, B and C. Now, if we join these points, then the sides of this triangular piece will be:
   (a) AB and BC only
   (b) AC and BC only
   (c) AB and AC only
   (d) AB, BC and AC
   Correct Option: (d) AB, BC and AC
   Solution:
   
   Naming the points as A, B and C.
   ABC is a triangle.
   Sides of this triangle are AB, BC and AC.
3. Danny owns an agricultural land which looks like a triangle when the corners of the land are connected with straight lines. He desires a seat in the centre where he can observe the entire area. For this, Danny take midpoint (say M) on one of the sides (say AB) of this land and join this midpoint with the corner point not on AB (say C). Then CM will be a/an
   
   
   (a) Altitude of a triangle
   (b) Median of a triangle
   (c) Midpoint of a triangle
   (d) Side of a triangle
   Correct Option: (b) Median of a triangle
   Solution:
   Danny come across with the following figure as per the given information:
   
   Here, ABC is a triangle and let’s join M and C as shown above.
   Since, M is the midpoint of AB and CM joins the midpoint of side AB and the vertex C.
   So, CM become the median of the triangle ABC.
4. There are multiple photo frames mounted on a wall.
   
   If the height of the top photo frame with corners points P, Q and R is PM, then PM is the
   (a) Median of triangle PQR
   (b) Altitude of triangle PQR
   (c) Side of triangle PQR
   (d) Bisector of side QR
   Correct Option: (b) Altitude of triangle PQR
   Solution:
   PM has one end point at a vertex of triangle PQR and other on the line containing the opposite side.
   Also, since PM is the height, so it is perpendicular to QR.
   
   Hence, PM is the altitude of triangle PQR.
5. Due to heavy rain, a traffic jam has been occurred in one of the part of a city. The three vehicles – a car, a bus and a motorbike are standing in such a way that they form a triangle (say XYZ).
   Consider the car’s position as XY, bus’s position as YZ and motorbike’s as XZ. When the car move forward, an exterior angle is formed as shown:
   
   
   Then the measure of ∠XZY is _____________.
   (a) 58°
   (b) 68°
   (c) 54°
   (d) 180°
   Correct Option: (c) 54°
   Solution:
   Here, ∠ZYM is an exterior angle of ∆XYZ.
   We know that the measure of an exterior angles is equal to the sum of its opposite interior angles.
   ⇒ ∠ZYM = ∠ZXY + ∠XZY
   ⇒ 112° = 58° + ∠XZY
   ⇒ 112° – 58° = ∠XZY
   ⇒ ∠XZY = 54°
6. The "Mile-a-minute Vine" is one of numerous plants having triangular leaves. It is an annual herbaceous vine called mile-a-minute weed. Its alternating, light green leaves range in size from 4 to 7 cm long to 5 to 9 cm wide, and have an equilateral triangular shape.
   
   What might the measures of the three angles be on any given leaf of this plant?
   (a) 50°, 60° and 70°
   (b) 45°, 45° and 90°
   (c) 30°, 60° and 90°
   (d) 60°, 60° and 60°
   Correct Option: (d) 60°, 60° and 60°
   Solution:
   As the leaves are in the shape of an equilateral triangle.
   So, all the three angles formed in every leave will be equal.
   Let measure of each angle be x°.
   ⇒ x° + x° + x° = 180°
   ⇒ 3x° = 180°
   ⇒ x° = 60°
   So, the measures of the three angles on any given leaf of this plant will be 60°, 60° and 60°.
7. Neeta has a cavity, therefore her mother advised her to use neem chew sticks rather than a toothbrush. She purchased 12 neem sticks of different sizes.
   
   
   These sticks have been separated into four groups of three sticks, and each group's sticks lengths are as follows:
   Group A: 3 cm, 4 cm, 5 cm
   Group B: 6 cm, 8 cm, 10 cm
   Group C: 8 cm, 15 cm, 23 cm
   Group D: 5 cm, 12 cm, 13 cm
   Which of the following group does not form a right angled triangle?
   (a) Group A
   (b) Group B
   (c) Group C
   (d) Group D
   Correct Option: (c) Group C
   Solution:
   To check whether the lengths of the sides form a right angled triangle, find the sum of the squares of the two smaller lengths and then check whether the sum is equal to the square of the longest side length.
   3² + 4² = 9 + 16 = 25 = 5²
   So, the sticks of lengths 3 cm, 4 cm, 5 cm form a right angled triangle.
   6² + 8² = 36 + 64 = 100 = 10²
   So, the sticks of lengths 6 cm, 8 cm, 10 cm form a right angled triangle.
   8² + 15² = 64 + 225 = 289 ≠ 23²
   So, the sticks of lengths 8 cm, 15 cm, 23 cm does form a right angled triangle.
   5² + 12² = 25 + 144 = 169 = 13²
   So, the sticks of lengths 5 cm, 12 cm, 13 cm form a right angled triangle.
   So, group C does not form a right angled triangle.
8. There is an emergency staircase on every story of the building. Its foot is 9 feet from the wall, while its peak or top is 12 feet above the floor. What may the staircase's possible length be if there are no twists and it is straight?
   
   
   (a) 3 feet
   (b) Approximately 7 feet
   (c) 14 feet
   (d) 15 feet
   Correct Option: (d) 15 feet
   Solution:
   Let points AB be the height of the staircase and BC be its distance from the wall.
   So, ABC forms a right angled triangle.
   
   
   By Pythagoras theorem, we have
   AC² = AB²+ BC²
   ⇒ AC² = (12)² + (9)²
   ⇒ AC²= 144 + 81 = 225
   ⇒ AC = 15 feet
   So, the height of the staircase is 15 feet.
9. Aditya, Raj and Sachin are enjoying ice-cream candies while returning from school. After they finished eating, they make a triangle using the sticks left. Aditya and Raj says that the lengths of the sticks they hold are 3 cm and 5 cm. What is the possible length of the stick that Sachin holds?
   
   
   (a) 2 cm
   (b) 4 cm
   (c) 8 cm
   (d) 9 cm
   Correct Option: (b) 4 cm
   Solution:
   We know that the sum of the lengths of any two sides of a triangle is greater than the third side.
   Here, 3 + 5 = 8
   So, the length of the third side must be less than 8 cm.
   Also, difference of the lengths of any two sides of a triangle is less than the third side.
   Difference = 5 – 3 = 2
   So, the length of third side must be greater 2 cm and less than 8 cm.
   Possible length of the stick that Sachin holds is 4cm.
   
   Subjective Questions
   
10. Rajiv used a triangular mould to cut a piece from a clay. He then measures the sides of that triangular piece and found that it is an equilateral triangle. Now keeping the clay stable, he drew one of the medians and then an altitude with the same vertex.
    
    
    State whether the medians of this triangular piece will coincide with its altitudes. Justify your answer.
    Solution:
    An equilateral triangle has all three sides equal and all three angles equal.
    So, a perpendicular from a vertex on the opposite side i.e. an altitude will bisect it. Thus, it become the median of the triangle.
    
    
    Thus, all three medians will coincide with the altitudes.
11. Shweta along with her family went for a picnic to Lonavala. While returning back, she got into a shop where she could find various hand bags. Shweta found a bag which was very attractive because of its triangular shape. She wanted to check if it can fit in her backpack and so took the rough measurements. If the measurements are as follows:
    
    
    Find the measure of ∠PRQ.
    Solution:
    Here, ∠PQX is an exterior angle of ∆PQR.
    We know that the measure of an exterior angles is equal to the sum of its opposite interior angles.
    ⇒ ∠PQX = ∠QPR + ∠PRQ
    ⇒ 120° = 50° + ∠PRQ
    ⇒ 120° – 50° = ∠PRQ
    ⇒ ∠PRQ = 70°
12. In order to save space, a triangular table is kept in one of the hall's corners. Now, the two sides of this table are touching the two adjacent walls, and these sides are perpendicular to one another. Find the measure of the two angles that are not in the middle of the two walls if the table's sides that touch the walls are equal.
    
    
    

    Solution:
    As the two walls are perpendicular to each other and the two sides of the table are touching them.
    So, the two sides of this table are perpendicular.
    That means, one of the angles of the triangle formed by the table is right angled.
    Also, the sides touching the walls are equal.
    So, it is an isosceles triangle and so the angles other than the right angle are equal.
    Let measure of each equal angle be x°.
    ⇒ 90° + x° + x° = 180°
    ⇒ 2x° = 180° – 90°
    ⇒ x° = 45°
    Thus, the measure of the remaining two angles are 45° and 45°.

13. Rahul entered the school from gate no. 1 and went to his class following a straight line. After the class was over, he first went to canteen and then took an exit from gate no. 1. The path took by him is shown below:
    
    
    Find the value of x and then the measures of the three angles.
    Solution:
    Here, ABC is a triangle and we now that sum of all three interior angles of a triangle is 180°.
    ⇒ (x – 3)° + (2x – 19)° + (x + 2)° = 180°
    ⇒ x – 3 + 2x – 19 + x + 2 = 180
    ⇒ 4x – 20 = 180
    ⇒ 4x = 200
    ⇒ x = 50
    ⇒ x – 3 = 47, 2x – 19 = 81, x + 2 = 52
    So, the three angles are 47°, 81° and 52°.
14. A worm fell into a paint box and Tarun assists the worm to get out of the box. The worm then begin to crawl on the floor, forming an irregular shape (shown below). Now, if the corner points are connected by a straight line, a triangle is formed with the following measurements as seen below:
    
    
    Based on the data given in the above figure, find the value of x.
    Solution:
    In the triangle, the two sides are equal to each other.
    So, it is an isosceles triangles and thus, the angles opposite to the equal sides are equal.
    ⇒ x° = 40°
    ⇒ x = 40
15. Gauri’s brother is in 4th standard and so he uses crayons for coloring the drawings made in the drawing book. There are just three crayons of different colors and sizes left and the others have been exhausted. The lengths of these crayons are 2 cm, 3 cm and 5cm.
    
    
    Now, if Gauri tries to create a triangle with these crayons, will she succeed? State the reason.
    Solution:
    Lengths of the three crayons are 2 cm, 3 cm and 5cm.
    For a triangle, sum of the lengths of any two sides of a triangle is greater than the length of the third side.
    But, 2 + 3 = 5 ≯ 5 (3rd side)
    So, a triangle with side lengths 2 cm, 3 cm and 5 cm cannot be created.
    Hence, Gauri will not succeed.
16. Three children Arjun, Rashmi and Suhani, are playing at the different corners of a park. Arjun and Rashmi are separated by 4.5 cm, Rashmi and Suhani by 5.8 cm, while Arjun and Suhani are separated by 10.2 cm. If the positions of these children are considered to be A, R and S respectively, state whether the triangle ARS may be created. Justify your answer.
    
    
    Solution:
    For a triangle, sum of the lengths of any two sides of a triangle is greater than the length of the third side.
    Distances are given as 4.5 cm, 5.8 cm 10.2 cm.
    4.5 + 5.8 = 10.3 > 10.2 (3rd side)
    4.5 + 10.2 = 14.7 > 5.8 (3rd side)
    5.8 + 10.2 = 16 > 4.5 (3rd side)
    So, the triangle with its side lengths 4.5 cm, 5.8 cm 10.2 cm is possible.
    Thus, the triangle ARS can be created.
17. Harsh is trying to make a sailing boat using a few sticks not necessarily of different lengths.
    
    
    He takes two sticks that are 5 cm and 12 cm long and arranges them so that they are perpendicular to one other. If the third stick he chooses has its length equal to the length of one of the two sticks picked earlier, will he be able to create a sail for the sailing board?
    Solution:
    Length of the two sticks are 5 cm and 12 cm.
    The edges (sides) of the sail make a right angled triangle.
    So, if the length of the third stick is 5 cm.
    5² + 12² = 25 + 144 = 169 ≠ 5²
    When the length of the third stick is 12 cm.
    5² + 12² = 25 + 144 = 169 ≠ 12²
    Thus, Harsh will not be able to create a sail for the sailing board.
    
    
18. There are multiple triangle-shaped chips in a packet, all with different side lengths. John had gone through more over half of the chips, leaving only four in the packet.
    
    
    He then basically stated that only two of the four chips form a right-angled triangle from observing the chips. These four chips' side lengths are listed below:
    3 cm, 4 cm, 5 cm
    1.2 cm, 3.5 cm, 3.7 cm
    1.6 cm, 6.3 cm, 6.5 cm
    5 cm, 7 cm, 13 cm
    Is John correct? Justify your answer.
    Solution:
    Consider the side lengths 3 cm, 4 cm, 5 cm
    3² + 4² = 25 = 5²
    So, they form a right angled triangle.
    Consider the side lengths 1.2 cm, 3.5 cm, 3.7 cm
    (1.2)² + (3.5)² = 1.44 + 12.25 = 13.69 = (3.7)²
    So, they form a right angled triangle
    Similarly, (1.6)² + (6.3)² = 2.56 + 39.69 = 42.25 = (6.5)²
    But, 5² + 7² = 25 + 49 = 74 ≠ (13)²
    Thus, three out of four chips form a right-angled triangle.
    Hence, John is incorrect.
19. The two flaps of a roof are designed to be perpendicular to one another. The flaps have lengths of 9 and 12 metres. Is it possible to join these flaps together at their ends with a thread that is 14 metres long? If not, suggest a possible length.
    Solution:
    The lengths of the two flaps are 9 m and 12 m and the flaps are perpendicular.
    Consider,
    9² + 12² = 81 + 144 = 225 ≠ 14²
    So, it is not possible to join these flaps together at their ends with a 14 m long thread.
    Also, 9² + 12² = 225 = 15²
    Thus, the length of the thread must be 15 m or more.
20. Ajay went for camping along with his parents. They attempt to construct a tent out of bamboos of various lengths. To do this, first make a right-angled triangle using three bamboos, and then make another right-angled triangle using the same vertical bamboo of previous triangle. The similar shape must be created on the opposite side of these triangles.
    
    
    If the lengths of the first three bamboos are 8 cm, 15 cm and 18 cm, will they be successful in the first step?
    Solution:
    The triangle created in the first step must be right angled triangle.
    Now, 8² + 15² = 64 + 225 = 289 = 17²
    So, the lengths of the first three bamboos must be 8 cm, 15 cm and 17 cm.
    Thus, Ajay and his parents will not succeed in the first step using these bamboos.