AP Board 7th Class Maths Solutions Chapter 13 Area and Perimeter Ex 3

AP State Syllabus AP Board 7th Class Maths Solutions Chapter 13 Area and Perimeter Ex 3 Textbook Questions and Answers.

AP State Syllabus 7th Class Maths Solutions 13th Lesson Area and Perimeter Exercise 3

Question 1.
Find the area of each of the following triangles.

Solution:
î) Area = [latex]\frac{1}{2}[/latex] = [latex]\frac{1}{2}[/latex] x 5 x 8 = 20 cm²
ii) Area = [latex]\frac{1}{2}[/latex] = [latex]\frac{1}{2}[/latex] x 6 x 4 = 12 cm²
iii) Area = [latex]\frac{1}{2}[/latex] = [latex]\frac{1}{2}[/latex] x 5.4 x 7.5 = 20.25 cm²
iv)Area = [latex]\frac{1}{2}[/latex] = [latex]\frac{1}{2}[/latex] x 6 x 4 = 12 cm²

Question 2.
In ΔPQR, PQ = 4 cm, PR =8 cm and RT = 6cm. Find (i) the area of ΔPQR (ii) the length of QS.

Solution:
Given PQ 4cm, PR = 8cm, RT = 6cm
1) Area of ΔPQR = [latex]\frac { 1 }{ 2 }[/latex] × base × height = [latex]\frac { 1 }{ 2 }[/latex] × PQ × RT
= [latex]\frac { 1 }{ 2 }[/latex] × 4 × 6 = 12cm²

ii) Also area of ΔPQR = [latex]\frac { 1 }{ 2 }[/latex] × base x height = [latex]\frac { 1 }{ 2 }[/latex] × PR × QS
12 = [latex]\frac { 1 }{ 2 }[/latex] × 8 × QS
QR = [latex]\frac{12 \times 2}{8}[/latex] = 3cm

Question 3.
ΔABC is right-angled at A. AD is perpendicular to BC, AB =5 cm, BC = 13 cm and AC = 12 cm. Find the area of ΔABC. Also, find the length of AD.

Solution:
Given.
In ΔABC, ∠A = 90°
AB = 5cm; AC = 12cm; AD ⊥ BC; BC = 13cm.
Now area of ΔABC = [latex]\frac { 1 }{ 2 }[/latex] × base × height
= [latex]\frac { 1 }{ 2 }[/latex] x AB x AC = [latex]\frac { 1 }{ 2 }[/latex] × 5 × 12 = 30cm²
Also area of MBC = [latex]\frac { 1 }{ 2 }[/latex] × BC × AD
30 = [latex]\frac { 1 }{ 2 }[/latex] × 13 × AD
∴ [latex]\frac{30 \times 2}{13}=\frac{60}{13}=4 \frac{8}{13}[/latex] cm

Question 4.
ΔPQR is isosceles with PQ = PR = 7.5 cm and QR = 9 cm. The height PS from P to QR, is 6 cm. Find the area of ΔPQR. What will be the height from R to PQ i.e. RT?

Solution:
Given: PQ = PR = 7.5cm, QR = 9cm PS = 6cm
Area of ΔPQR = [latex]\frac { 1 }{ 2 }[/latex] bh = latex]\frac { 1 }{ 2 }[/latex] x 9 x 6 = 27cm²
Also area of ΔPQR = [latex]\frac { 1 }{ 2 }[/latex] × PQ × RT
27 = [latex]\frac { 1 }{ 2 }[/latex] × 7.5 × RT [PQ = PR]
∴ RT = [latex]\frac{2 \times 27}{7.5}=\frac{2 \times 27 \times 2}{15}=\frac{36}{5}[/latex] = 7.2 cm

Question 5.
ABCD rectangle with AB =8 cm, BC = 16 cm and AE = 4 cm. Find the area of ABCE. Is the area of ΔBEC equal to the sum of the area of ΔBAE and ΔCDE. Why?

Solution:
Given : In rectangle ABCD,
AB = 8cm, BC = 16 cm, AE = 4cm
Now Area of ΔBAE = [latex]\frac { 1 }{ 2 }[/latex] x base x height
= [latex]\frac { 1 }{ 2 }[/latex] × [latex]\overline{\mathrm{BC}} \times \overline{\mathrm{AB}}[/latex]
= [latex]\frac { 1 }{ 2 }[/latex] × 16 x 8 = 64cm²

Area of ΔCDE = [latex]\frac { 1 }{ 2 }[/latex] x base x height
= [latex]\frac { 1 }{ 2 }[/latex] × [latex]\overline{\mathrm{ED}} \times \overline{\mathrm{DC}}[/latex]
= [latex]\frac { 1 }{ 2 }[/latex] × 12 × 8 = 48cm²
(∵ [latex]\overline{\mathrm{ED}}[/latex] = AD – AE = 16 – 4 = 12cm and [latex]\overline{\mathrm{DC}}=\overline{\mathrm{AB}}[/latex])
Now ΔBAE + ΔCDE = 16 + 48 = 64 = ΔBCE.

Question 6.
Ramu says that the area of ΔPQR is, A = [latex]\frac{1}{2}[/latex] × 7 × 5 cm².
Gopi says that it is, A = [latex]\frac{1}{2}[/latex] × 8 × 5 cm². Who is correct’? Why?

Solution:
Ramu is correct.
Since area of a triangle = [latex]\frac { 1 }{ 2 }[/latex] × base × corresponding height
= [latex]\frac { 1 }{ 2 }[/latex] × 7 × 5cm²
But height 5cm is not corresponding to side &m.
∴ Gopi is not correct.

Question 7.
Find the base of a triangle whose area is 220 cm² and height is 11cm.
Solution:
Given Area = 220 cm², h = 11 cm
Area of a triangle = [latex]\frac { 1 }{ 2 }[/latex]b.h
= [latex]\frac { 1 }{ 2 }[/latex] × b × 11 = 220cm² (given)
b × 11 = 220 × 2
b = [latex]\frac{220 \times 2}{11}[/latex] = 40
∴ base of the triangle = 40 cm.

Question 8.
In a triangle the height is double the base and the area is 400 cm². Find the length of the base and height.
Solution:
Area = 400 cm²
Let the base of the triangle be x.
height = 2x
Area of the triangle = [latex]\frac { 1 }{ 2 }[/latex]b.h
[latex]\frac { 1 }{ 2 }[/latex] × x × 2x = 400 (given)
x² = 400
x = [latex]\sqrt{400}[/latex] = 20
∴ base of the triangle = 20 cm
height = 2(base)= 2(20) = 40 cm.

Question 9.
The area of triangle is equal to the area of a rectangle whose length and breadth are 20 cm and 15 cm respectively. Calculate the height of the triangle if its base measures 30 cm.
Solution:
Given :Length of the rectangle = 20 cm
Breadth of the rectangle = 15 cm
Base of the triangle = 30 cm
Area of the rectangle = Area of the triangle
Length x breadth = [latex]\frac { 1 }{ 2 }[/latex] × base × height
20 × 15 = [latex]\frac { 1 }{ 2 }[/latex] × 30 × height
∴ height = [latex]\frac{20 \times 15}{15}[/latex] = 20cm

Question 10.
In Figure ABCD find the area of the shaded region.

Solution:
Given : Side of the square ABCD = 40cm
Area of the shaded rgion = (Area of the square) — (Area of the unshaded region)
= (side × side) – [latex]\frac { 1 }{ 2 }[/latex] × base × height
= 40 × 40 – [latex]\frac { 1 }{ 2 }[/latex] × 40 × 40
= 1600 – 800 = 800cm²

Question 11.
In Figure ABCD, find the area of the shaded region.

Solution:
Given: Side of the square = 20 cm
Area of the shade region
= (Area of the square) – (Area of unshaded region)
= (Square ABCD) – (ΔEAF + ΔFBC + ΔEDC)
= side × side – ([latex]\frac { 1 }{ 2 }[/latex]AF × AE + [latex]\frac { 1 }{ 2 }[/latex] FB × BC + [latex]\frac { 1 }{ 2 }[/latex] DE × DC)
= 20 × 20 – ([latex]\frac { 1 }{ 2 }[/latex] × 10 × 12 + [latex]\frac { 1 }{ 2 }[/latex] × 10 × 20 + [latex]\frac { 1 }{ 2 }[/latex] × 8 × 20)
= 400 – [60 + 100 + 80] = 400 – 240 = 160cm²

Question 12.
Find the area of a parallelogram PQRS, if PR =24 cm and QU = ST =8 cm.

Solution:
Given : In PQRS, PR = 24cm, ST = 8cm, QU =8cm
Area of PQRS = ΔPQR + ΔPRS
= [latex]\frac { 1 }{ 2 }[/latex] × [latex]\frac { 1 }{ 2 }[/latex]base x height + x base x height
= [latex]\frac { 1 }{ 2 }[/latex] × 24 x 8 + [latex]\frac { 1 }{ 2 }[/latex] × 24 × 8 = 96 + 96 = 192 cm²

Question 13.
The base and height of the triangle are in the ratio 3:2 and its area is 108 cm². Find its base
and height.
Solution:
Given : Area = 108 cm2
Let the base of the triangle be 3x and height of the triangle be 2x.
Area of the triangle = [latex]\frac { 1 }{ 2 }[/latex] × base × height = [latex]\frac { 1 }{ 2 }[/latex] x (3x) (2x) = 108 cm² (given)
3x² = 108
x² = 108/3 = 36
x = [latex]\sqrt{36}[/latex] = 6
∴ base 3x = 3(6) = 18 cm
height = 2x = 2(6) = 12cm