# Properties of Triangles

## Exercise 6.5

**Question 1:** PQR is a triangle, right angled at P. If PQ = 10 cm and PR = 24 cm, find QR.

**Answer:** Using Pythagoras rule,

*QR ^{2} = PQ^{2} + PR^{2}*

Or, *QR ^{2} = 10^{2} + 24^{2} = 100 + 576 = 676*

Or, *QR ^{2} = 2 × 2 × 13 × 13*

Or, *QR = 2 × 13 = 26* cm

**Question 2:** ABC is a triangle, right angled at C. If AB = 25 cm and AC = 7 cm, find BC.

**Answer:** Using Pythagoras rule,

`AB^2 = AC^2 + BC^2`

Or, `25^2 = 7^2 + BC^2`

Or, `625 = 49 + BC^2`

Or, `BC^2 = 625 – 49 = 576`

Or, `BC^2 = 2 xx 2 xx 2 xx 2 xx 2 xx 2 xx 3 xx 3 ``= 2^2 xx 2^2 xx 2^2 xx 3^2`

Or, `BC = 2 xx 2 xx 2 xx 3 = 24`

**Question 3:** A 15 m long ladder reached a window 12 m high from the ground on placing it against a wall at a distance a. Find the distance of the foot of the ladder from the wall.

**Answer:** The ladder is making hypotenuse while the wall is making one of the legs of the triangle.

Using Pythagoras rule, `h^2 = p^2 + b^2`

Or, `b^2 = h^2 – p^2 = 15^2 – 12^2`

Or, `b^2 = 225 – 144 = 81`

Or, `b^2 = 3 xx 3 xx 3 xx 3 = 3^2 xx 3^2`

Or, `b = 3 xx 3 = 9`

**Question 4:** Which of the following can be the sides of a right triangle?

(a) 2.5 cm, 6.5 cm, 6 cm

**Answer:** Square of the longest side `= 6.5^2 = 42.25`

Now, `2.5^2 = 6.25` and `6^2 = 36`

Adding above two, we get; `6.25 + 36 = 42.25` which is equal to the square of the longest side.

Hence, it is a right angled triangle.

(b) 2 cm, 2 cm, 5 cm

**Answer:** Square of the longest side `= 5^2 = 25`

Now, `2^2 = 4`

Adding the squares of remaining two sides, we get, `4 + 4 = 16`

This figure is not equal to the square of the longest side.

Hence, this is not a right angled triangle.

(c) 1.5 cm, 2 cm, 2.5 cm

**Answer:** Square of the longest side `= 2.5^2 = 6.25`

Now, `1.5^2 = 2.25` and `2^2 = 4`

Adding the above two, we get, `2.25 + 4 = 6.25`

This figure is equal to the square of the longest side.

Hence, it is a right angled triangle.

In case of right angled triangles, identify the right angles.

**Answer:** In options ‘a’ and ‘c’, right angle is made by the smaller sides or legs of the triangle.

**Question 5:** A tree is broken at a height of 5 m from the ground and its top touches the ground at a distance of 12 m from the base of the tree. Find the original height of the tree.

**Answer:** We have; two legs of right angled triangle = 5 m and 12 m

Hypotenuse can be calculated as follows:

`h^2 = p^2 + b^2`

Or, `h^2 = 5^2 +12^2 ``= 25 + 144 = 169`

Or, `h^2 = 13 xx 13`

Or, `h = 13` m

So, original height of tree `= 12 + 13 = 25` m

**Question 6:** Angles Q and R of a ΔPQR are 25° and 65°. Write which of the following is true.

(a) `PQ^2 + QR^2 = RP^2`

(b) `PQ^2 + RP^2 = QR^2`

(c) `RP^2 + QR^2 = PQ^2`

**Answer:** Here, Sum of angles Q and R `= 25° + 65° = 90°`

So, angle `P = 90°`

This means, `h = QR` while other sides are PQ and PR.

So, `QR^2 = PQ^2 + PR^2`

Hence, option ‘b’ is correct.

**Question 7:** Find the perimeter of a rectangle whose length is 40 cm and a diagonal is 41 cm.

**Answer:** Breadth of the rectangle can be calculated by using Pythagoras rule because length, breadth and diagonal would make a right angled triangle.

So, `h^2 = p^2 + b^2`

Or, `41^2 = 40^2 + b^2`

Or, `1681 = 1600 + b^2`

Or, `b^2 = 1681 – 1600 = 81`

Or, `b^2 = 9 xx 9`

Or, `b = 9` cm.

Perimeter can be calculated as follows:

Perimeter = 2(length + breadth)

`= 2(40 + 9) = 2 × 49 = 98` cm

**Question 8:** The diagonals of a rhombus measure 16 cm and 30 cm. Find its perimeter.

**Answer:** Halves of diagonals of a rhombus make the legs of a right angle triangle while hypotenuse is made by a side of the rhombus. So, side of the rhombus can be calculated by using Pythagoras rule;

We know, `h^2 = p^2 + b^2`

Or, `h^2 = 8^2 + 15^2 ``= 64 + 225 = 289`

Or, `h^2 = 17 xx 17`

Or, `h = 17` cm

Hence, Perimeter `= 17 × 4 = 68` cm