# Properties of Triangles

## In Text Questions

**Example 1:** How many altitudes can a triangle have?

**Answer:** Three

**Example 2:** Draw rough sketches of altitudes from A to BC for the following triangles.

**Answer:**

**Example 3:** Will an altitude always lie in the interior of a triangle? If you think that this need not be true, draw a rough sketch to show such a case.

**Answer:** Answer of previous question shows obtuse angle triangle in which one of the altitudes will be outside the triangle.

**Example 4:** Can you think of a triangle in which two altitudes of the triangle are two of its sides?

**Answer:** Right angled triangle

**Example 5:** Can the altitude and median be same for a triangle?

**Answer:** This can happen in equilateral triangle and in isosceles triangle.

**Example 6:** Find x in each figure:

**Answer:** (i) `x = 40°`

Because, angles opposite to equal sides of a triangle are equal.

**Answer:** (ii) In this case, the third angle = 45°

Reason is same as in previous question.

Hence, `x = 180° - (45° + 45°) ``= 180° - 90°= 90°`

**Answer:** (iii) `x = 50°` because of the same reason as in first question.

**Answer:** (iv) Here; `100° + x + x = 180°`

Or, `100° + 2x = 180°`

Or, `2x = 180° - 100° = 80°`

Or, `x = 80° ÷ 2 = 40°`

**Answer:** (v) This is a right angled triangle in which both the legs are equal. So, both the angles (other than right angle) are equal. Hence, `x = 45°`

**Answer:** (vi) Here, x is equal to the third angle because this is an isosceles triangle.

Hence, `40° + 2x = 180°`

Or, `2x = 180° - 40° = 140°`

Or, `x = 140° ÷ 2 = 70°`

**Answer:** (vii) Here, angle opposite to x is equal because it is an isosceles triangle.

Or, `x + 120° = 180°`

Or, `x = 180° - 120° = 60°`

**Answer:** (viii) Sum of internal angles which are opposite to external angle = 110°

Both the internal angles (opposite to external angle) are equal because it is an isosceles triangle.

Hence, `2x = 110°`

Or, `x = 110° ÷ 2 = 55°`

**Answer:** (ix) Angle opposite to x is 30° because vertically opposite angles are equal.

As this is an isosceles triangle, hence, `x = 30°`

**Example 7:** Find the value of x in following figures:

**Answer:** (i) `x + y = 120°` (External angle is equal to sum of opposite internal angles.)

The third angle of triangle = 180° - 120° = 60°(Linear pair of angles)

Now, y = 60° (Angles opposite to equal sides).

Now, `x + y = 120°`

Or, `x + 60° = 120°`

Or, `x = 120° - 60°`

**Answer:** (ii) Both the internal angles of given triangle (other than right angle) are equal and hence = 45°

Hence, `y = 180° - 45° = 135°`

**Answer:** (iii) The third angle in this triangle = 92° (Vertically opposite angles)

Remaining two angles will be equal because they are opposite the equal sides.

Hence, `x + x + 92° = 180°`

Or, `2x = 180° - 92° = 88°`

Or, `x = 88° ÷ 2 = 44°`

**Example 8:** Find the unknown length x in the following figures:

**Answer:** (i) Measure of two legs of right angled triangle = 3 and 4

Using Pythagoras rule,

`x^2 = 3^2 + 4^2 = 9 + 16 = 25`

Or, `x^2 = 5 xx 5`

Or, `x = 5`

**Answer:** (ii) Measure of two legs of right angled triangle = 6 and 8

Using Pythagoras rule,

`x^2 = 6^2 + 8^2 = 36 + 64 = 100`

Or, `x^2 = 10 xx 10`

Or, `x = 10`

**Answer:** (iii) Measure of two legs of right angled triangle = 8 cm and 15 cm

Using Pythagoras rule,

`x^2 = 8^2 + 15^2 ``= 64 + 225 = 289`

Or, `x^2 = 17 xx 17`

Or, `x = 17`

**Answer:** (iv) Measure of two legs of right angled triangle = 24 and 7

Using Pythagoras rule,

`x^2 = 24^2 + 7^2 ``= 576 + 49 = 625`

Or, `x^2 = 5^2 xx 5^2`

Or, `x = 5 xx 5 = 25`

**Answer:** (v) In this case, half of x can be calculated by using 37 as hypotenuse and 12 as perpendicular of one of the two right angled triangles in the figure.

`b^2 = h^2 – p^2`

Or, `b^2 = 37^2 – 12^2`` = 1369 – 144 = 1225`

Or, `b^2= 5^2 xx 7^2`

Or, `b = 5 xx 7 = 35`

Hence, `x = 2 xx 35 = 70`

**Answer:**(vi) For the smaller triangle on the right side, h = 5 cm and p = 3 cm

Hence, `b^2 = h^2 – p^2`

Or, `b^2 = 5^2 – 3^2 = 25 – 9 = 16`

Or, `b = 4` cm

Now, for the larger triangle, `p = 12` cm and `b = 5` cm

Hence, `h^2 = 12^2+ 5^2`

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

Or, `h = 13`

Value of `x = 13 – 4 = 9` cm