Page 179 - ICSE Math 7

Example 3: For the adjoining figure, find angles x, y and z. P
Solution: ∠QTP + ∠QTR = 180° (Linear pair) S 55° T

⇒ ∠QTP + 90° = 180° U
⇒ ∠QTP = 180° – 90° = 90° x y z
Similarly, ∠RSP = 90° Q R
In DPQT,
∠PQT + ∠QTP + ∠TPQ = 180° (Sum of the angles of a triangle)
⇒ x + 90° + 55° = 180°

⇒ x = 180° – 145° = 35°
In DPRS,
∠PRS + ∠RSP + ∠SPR = 180° (Sum of the angles of a triangle)
⇒ z + 90° + 55° = 180°
⇒ z = 180° – 145° = 35°
In DSQU,
∠QUR = ∠UQS + ∠USQ (Exterior angle is equal to the sum of its
two interior opposite angles)
⇒ y = x + 90°
⇒ y = 35° + 90° = 125°

Example 4: The angles of a triangle are in the ratio 3 : 7 : 8. Find the angles.
Solution: Let the angles be 3x, 7x and 8x. Try This
As the sum of the angles of a triangle is 180°, therefore
3x + 7x + 8x = 180° If one angle of a triangle
⇒ 18x = 180° is equal to the sum of the
other two, then the triangle
180° is a right-angled triangle.
⇒ x = = 10°
18 Prove this statement.
\ 3x = 3 × 10° = 30°, 7x = 7 × 10° = 70° and 8x = 8 × 10° = 80°
Thus, the angles of the triangle are 30°, 70° and 80°.

Altitude and Median of a Triangle

The altitude of a triangle is the perpendicular
drawn from its vertex to its opposite side. In the P
adjoining figure, PS is the altitude of DPQR drawn Maths Info
from vertex P. A triangle has three altitudes and they Orthocentre of a triangle
are concurrent. Also, the point of concurrence of may lie inside or outside
altitudes is known as the orthocentre of a triangle. Q R the triangle.
The line segment joining the vertex of a triangle S
to the midpoint of its opposite side is known as a
median. A
In the adjoining figure, AD is the median of Maths Info
DABC drawn from vertex A. A triangle has three
medians and they are concurrent. Also, the point of Centroid of a triangle
always lies inside the
concurrence of medians is known as the centroid triangle.
of a triangle. B D C

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