Properties of Triangles NCERT 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