# Construct an isosceles triangle whose base is 8 cm and altitude 4 cm and then another triangle whose sides are 1 (1/2) times the corresponding sides of the isosceles triangle

**Solution:**

- Draw the line segment of the base 8 cm. Draw perpendicular bisector of the line. Mark a point on the bisector which measures 4 cm from the base. Connect this point from both ends.
- Then draw another line that makes an acute angle with the given line. Divide the line into m + n parts where m and n are the ratios given.
- Two triangles are said to be similar if their corresponding angles are equal. They are said to satisfy Angle-Angle-Angle (AAA) Axiom.
- The basic proportionality theorem states that “If a straight line is drawn parallel to a side of a triangle, then it divides the other two sides proportionally".

Steps of construction:

- Draw BC = Through D, the mid-point of BC, draw the perpendicular to BC and cut an arc from D on it such that DA = 4cm. Join BA and CA. ΔABC is obtained.
- Draw the ray BX so that ∠CBX is acute.
- Mark 3 (3 > 2 in 1(1/2) = 3/2) points B
_{1}, B_{2}, B_{3}on BX such that BB_{1}= B_{1}B_{2}= B_{2}B_{3} - Join B
_{2}(2^{nd}point ∵ 2 < 3) to C and draw B_{3}C' parallel to B_{2}C , intersect BC extended at C’. - Through C’ draw C'A' parallel to CA to intersect BA extended to A’. Now, ΔA'BC' is the required triangle similar to ΔABC where BA'/BA = C'A'/CA = BC'/BC = 3/2

Proof:

In ΔBB_{3}C', B_{2}C || B3C',

Hence by Basic proportionality theorem,

B_{2}B_{3}/BB_{2} = CC'/BC = 1/2

Adding 1,

CC'/BC + 1 = 1/2 + 1

(BC+CC')/BC = 3/2

BC'/BC = 3/2

Consider ΔBAC and ΔBA'C'

∠ABC = ∠A'BC' (Common)

∠BCA = ∠BC'A' (Corresponding angles ∵ CA || C'A')

∠BAC = ∠BA'C' (Corresponding angles)

By AAA axiom, ΔBAC ~ ΔBA'C'

∴ Corresponding sides are proportional

Hence,

BA'/BA = BC'/BC= CA'/CA = 3/2

**Video Solution:**

## Construct an isosceles triangle whose base is 8 cm and altitude 4 cm and then another triangle whose sides are 1 1/2 times the corresponding sides of the isosceles triangle

### NCERT Solutions Class 10 Maths - Chapter 11 Exercise 11.1 Question 4:

Construct an isosceles triangle whose base is 8 cm and altitude 4 cm and then another triangle whose sides are 1 1/2 times the corresponding sides of the isosceles triangle

An isosceles triangle ABC whose base is 8 cm and altitude 4 cm and then another triangle A'BC' whose sides are 1 1/2 times the corresponding sides of the isosceles triangle have been constructed