Frank solutions for Mathematics [English] Class 9 ICSE chapter 17 - Pythagoras Theorem [Latest edition]

Find the length of the perpendicular of a triangle whose base is 5cm and the hypotenuse is 13cm. Also, find its area.

Find the length of the hypotenuse of a triangle whose other two sides are 24cm and 7cm.

Calculate the area of a right-angled triangle whose hypotenuse is 65cm and one side is 16cm.

A man goes 10 m due east and then 24 m due north. Find the distance from the straight point.

A ladder 25m long reaches a window of a building 20m above the ground. Determine the distance of the foot of the ladder from the building.

A right triangle has hypotenuse p cm and one side q cm. If p - q = 1, find the length of third side of the triangle.

A ladder 15m long reaches a window which is 9m above the ground on one side of a street. Keeping its foot at the same point, the ladder is turned to other side of the street to reach a window 12m high. Find the width of the street.

The foot of a ladder is 6m away from a wall and its top reaches a window 8m above the ground. If the ladder is shifted in such a way that its foot is 8m away from the wall to what height does its tip reach?

Two poles of height 9m and 14m stand on a plane ground. If the distance between their 12m, find the distance between their tops.

The length of the diagonals of rhombus are 24cm and 10cm. Find each side of the rhombus.

Each side of rhombus is 10cm. If one of its diagonals is 16cm, find the length of the other diagonals.

In ΔABC, AD is perpendicular to BC. Prove that: AB² + CD² = AC² + BD²

In an equilateral triangle ABC, the side BC is trisected at D. Prove that 9 AD² = 7 AB².

From a point O in the interior of aΔABC, perpendicular OD, OE and OF are drawn to the sides BC, CA and AB respectively. Prove that: AF² + BD² + CE² = OA² + OB² + OC² - OD² - OE² - OF²

From a point O in the interior of aΔABC, perpendicular OD, OE and OF are drawn to the sides BC, CA and AB respectively. Prove that: AF² + BD² + CE² = AE² + CD² + BF²

In a triangle ABC, AC > AB, D is the midpoint BC, and AE ⊥ BC. Prove that: AC² = AD² + BC x DE + `(1)/(4)"BC"^2`

In a triangle ABC, AC > AB, D is the midpoint BC, and AE ⊥ BC. Prove that: AB² = AD² - BC x CE + `(1)/(4)"BC"^2`

In a triangle ABC, AC > AB, D is the midpoint BC, and AE ⊥ BC. Prove that: AB² + AC² = 2AD² + `(1)/(2)"BC"^2`

In a triangle ABC, AC > AB, D is the midpoint BC, and AE ⊥ BC. Prove that: AC² - AB² = 2BC x ED

In a triangle ABC, AC > AB, D is the midpoint BC, and AE ⊥ BC. Prove that: AB² + AC² = 2(AD² + CD²)

A point OI in the interior of a rectangle ABCD is joined with each of the vertices A, B, C and D. Prove that  OB² + OD² = OC² + OA²

ABCD is a rhombus. Prove that AB² + BC² + CD² + DA²= AC² + BD²

AD is perpendicular to the side BC of an equilateral ΔABC. Prove that 4AD² = 3AB².

The perpendicular AD on the base BC of a ∆ABC intersects BC at D so that DB = 3 CD. Prove that `2"AB"^2 = 2"AC"^2 + "BC"^2`

In a triangle ABC right angled at C, P and Q are points of sides CA and CB respectively, which divide these sides the ratio 2 : 1.
Prove that: 9AQ² = 9AC² + 4BC² 

In a triangle ABC right angled at C, P and Q are points of sides CA and CB respectively, which divide these sides the ratio 2 : 1.
Prove that: 9BP² = 9BC² + 4AC²

In a triangle ABC right angled at C, P and Q are points of sides CA and CB respectively, which divide these sides the ratio 2 : 1.
Prove that : 9(AQ² + BP²) = 13AB² 

In the given figure, PQ = `"RS"/(3)` = 8cm, 3ST = 4QT = 48cm.
SHow that ∠RTP = 90°.

In a right-angled triangle ABC,ABC = 90°, AC = 10 cm, BC = 6 cm and BC produced to D such CD = 9 cm. Find the length of AD.

In the given figure. PQ = PS, P =R = 90°. RS = 20 cm and QR = 21 cm. Find the length of PQ correct to two decimal places.

In a right-angled triangle PQR, right-angled at Q, S and T are points on PQ and QR respectively such as PT = SR = 13 cm, QT = 5 cm and PS = TR. Find the length of PQ and PS.

PQR is an isosceles triangle with PQ = PR = 10 cm and QR = 12 cm. Find the length of the perpendicular from P to QR.

In a square PQRS of side 5 cm, A, B, C and D are points on sides PQ, QR, RS and SP respectively such as PA = PD = RB = RC = 2 cm. Prove that ABCD is a rectangle. Also, find the area and perimeter of the rectangle.