# Properties of Triangles

## In Text Questions

Example 1: How many altitudes can a triangle have?

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

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?

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 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