Class 9 Maths Chapter 10 HOTS Questions - Heron’s Formula

Q1: How much paper of each shade is needed to make a kite given in the figure, in which ABCD is a square with diagonal 60 cm?
Sol:
Since diagonals of a square are of equal length and bisect each other at right angles, therefore,
Area of ∆AOD = 1/2 × 30 × 30 = 450 cm²
Area of ∆AOD = Area of ∆DOC = Area of ∆BOC
= Area of ∆AOB = 450 cm2
[∵ ∆AOD = ∆AOB ≅ ∆BOC ≅ ∆COD, ∵ they have equal area]
Now, area of ACEF (by Heron’s formula)
Here a = 20 cm, b = 20 cm and c = 30 cm

Area of ∆CEF =

Now, area of orange shaded paper in kite
= Area of ∆AOD + Area of ∆CEF
= 450 cm² + 198.4 cm²
= 648.4 cm²
Area of blue shaded paper in kite
= Area of ∆AOB + Area of ∆COD
= 450 cm² + 450 cm² = 900 cm²
Area of black shaded paper in kite = Area of ∆BOC = 450 cm².

Q2: OPQR is a rhombus, whose three vertices P, Q and R lie on the circle with centre 0. If the radius of the circle is 12 cm, find the area of the rhombus.
Sol:
Since diagonals bisect each other at 90°.
∴ In right ∆QLR, (LR)² + (LQ)² = (QR)²


Area of rhombus OPQR = 1/2 x products of diagonals
=
=

Q3: In an exhibition, an umbrella is made by stiching 10 triangular pieces of cloth with same message written on two triangular pieces. If each piece of cloth measures 60 cm, 60 cm and 20 cm, find how much cloth is required for each message.
Why we should respect women, educate women, save women and empower women ?
Sol:
Here, each triangular piece is an isosceles triangle with sides 60 cm, 60 cm and 20 cm.
Area of each triangular Piece =

Now, there are 2 triangular pieces with same message.
∴ Total area of cloth for each message = 2 × 591.61 = 1183.22 cm²
We should respect women, educate women, save women and empower women to serve humanity and give them equal opportunity so far they deprived

Q4: Sister Nivedita has trapezium shaped plot which she divided into three triangular portion for different purposes. I – for providing free education for orphan children, II – for providing dispensary for the needy villagers and III – for the library for villagers. Find the area of trapezium plot given in the figure. Which qualities of sister Nivedita are being depicted in question ?
Sol:
Here, ABCD is the trapezium with AB || DC.
Through C, Draw CF ⊥ AB
For – I: For area of ∆EBC

=
= 336m²
But, we know, Area of ∆EBC = 1/2(base × height)
⇒ 1/21 × 30 × CF = 336
15 × CF = 336
⇒ CF = 336 / 15 = 22.4 m
Now, area of trapezium shaped plot = 1/2 (20 + 50)(22.4)
= 35 x 22.4 = 784 m²
Caring, kind, social, generous and visionary lady.