Question

In ΔABC, <ABC=90° and BO is parallel to AC. If AB=5.7cm, BO=3.8cm, CD=5.4cm. What is the length of BC?
a. 7.8 cm
b. 8.7 cm
c. 8 cm
d. 8.1 cm​

Answer 1

Triangles

The following is the concept and tips that will be used to find the solution:

• Corresponding angles are cingurent and corresponding sides of similar triangle are proportional.

We've been given that, In right ΔABC, ∠ABC = 90° and BO is parallel to AC. With this information, we've been asked to find out the length of BC if AB = 5.7cm, BO = 3.8cm, CD = 5.4cm.

In right ∆ABC and ∆BDC, we have;

• ∠ABC = ∠BDC [Each 90°]
• ∠BCA = ∠BCD [Common angles]

By AA similarity theorem, we get:

• ∆ABC ~ ∆BDC [AA similarity]

We know that, corresponding sides of similar triangle are proportional. So,

$\implies \dfrac{AB}{BD} = \dfrac{BC}{CD} \\ \\ \implies \dfrac{5.7}{3.8} = \dfrac{BC}{5.4} \\ \\ \implies 1.5 = \dfrac{BC}{5.4} \\ \\ \implies BC = 1.5 \times 5.4 \\ \\ \implies \boxed{ \bf{BC = 8.1}}$

Hence, the length of BC is 8.1cm. So, option (d) 8.1cm is correct.

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MORE TO KNOW

SSS theorem : SSS similarity theorem states that, If the three sides of one triangle are equal to the three sides of another triangle, the triangles are congruent.

SAS theorem : SAS similarity theorem states that, if the two sides and the angle between those two sides are equal to another triangle, the triangles are congruent.

AAA theorem : AAA similarity theorem states that, If the three angles of one triangle are equal to the three angles of another triangle, the triangles are congruent.

AA theorem : AA similarity theorem states that, If two angles of one triangle are equal to the two angles of another triangle, then the triangles are similar.

Answer 2

Answer:

Correct Question :-

✯ In ΔABC, <ABC=90° and BO is parallel to AC. If AB=5.7cm, BD =3.8cm, CD=5.4cm. What is the length of BC?

Options :

• ☯ a. 7.8 cm
• ☯ b. 8.7 cm
• ☯ c. 8 cm
• ☯ d. 8.1 cm

Given :

✯ In ΔABC, <ABC=90° and BO is parallel to AC. If AB=5.7cm, BD =3.8cm, CD=5.4cm.

Find Out :-

✯ What is the length of BC?

Solution :-

Firstly, we have to show that △ABC ~△BDC

⦿ Let △ABC and △BDC

⦿ ∠ABC = ∠BDC [each 90°]

⦿ ∠ACB = ∠BCD [common angle]

∴ △ABC ~ △BDC [by AA similarity criterion]

Since, triangles are similar, hence corresponding sides are proportional.

We have :

• AB = 5.7cm,
• BD = 3.8cm,
• CD = 5.4cm

So, according to the question or ATQ :-

$\sf \longrightarrow \sf \dfrac{AB}{BC} =\: \dfrac{BD}{DC}$

$\sf \longrightarrow \sf \dfrac{5.7}{BC} =\: \dfrac{3.8}{5.4}$

$\sf \longrightarrow \sf BC =\: \dfrac{5.7 \times 5.4}{3.8}$

$\sf \longrightarrow \sf BC =\: \dfrac{30.78}{3.8}$

$\longrightarrow$ ${\small{\bold{\purple{\underline{BC =\: 8.1\: cm}}}}}$

Henceforth, the length of BC is 8.1 cm.

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