Page 179 - Maths Skills - 7

The Triangle and Its Properties 177

8. In Fig., PQR is an isosceles triangle with PR = QR.

If ∠P = 80°, find

(i) ∠PQR and ∠PRQ, and

(ii) the values of ∠a and ∠b.

PYTHAGORAS’ THEOREM P
All right-angled triangles have a unique property related to them, which was
first identified by an Indian mathematician Baudhayana, but became famous Perpendicular Hypotenuse
when the Greek Philosopher Pythagoras provided the proof of this theorem
and hence it is known as Pythagoras Theorem. According to the Pythagoras Q R
Theorem, “In a right angled triangle, the square of the hypotenuse is equal Base
to the sum of the squares of the other two sides.”

For example, In ∆PQR, ∠Q = 90° and PR is the hypotenuse.
PR = PQ + QR or (Hypotenuse) = (Perpendicular) + (Base) 2
2
2
2
2
2
Fact-o-meter
� In a right-angled triangle, the hypotenuse is the longest side.
� Of all the line segments that can be drawn to a given line from a point outside it, the
perpendicular line segment is the shortest side.

Converse of Pythagoras Theorem

The converse of pythagoras Theorem states that: “If the square of one side of a triangle is equal to the sum of
the squares of the other two sides, then the triangle is a right-angled triangle with the A
angle opposite to the first side being right angle.”

For example, in ∆ACB, if AB = BC + CA , then the ∆ACB is a right-angled triangle,
2
2
2
right-angled at C.
Pythagorean Triplets C B
A set of three natural numbers a, b, c in this order forms a Pythagorean triplet if c = a + b . For example, each
2
2
2
of the triplets (3, 4, 5), (5, 12, 13) and (6, 8, 10) is a Pythagorean triplet.
Let’s Attempt

Example 1: Two buildings 30 m and 15 m high, stand upright on a ground. If they are 36 m apart, find the
distance between their tops. B
Solution: Let AB and CD represent the given buildings such that
AB = 30 m, CD = 15 m and AC = 36 m m

Join BD. From D, draw DE ^ AB. D E 30
m
Then, ED = AC = 36 m 15
BE = AB – AE = AB – CD = (30 – 15) m = 15 m C 36 m A