If sides of two similar triangles are in the ratio 8 is to 10

Solution:

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Sides of two similar triangles are in the ratio 4:9. Areas of these triangles are in the ratio (A) 2 : 3 (B) 4: 9 (C) 81: 16 (D) 16: 81
Corresponding Sides
Calculating the Lengths of Corresponding Sides
Step 1: Find the ratio
Step 2: Use the ratio

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides

Given that,

Sides of two similar triangles are in the ratio 4: 9.

We know that,

The ratio of the areas of two similar triangles = square of the ratio of their corresponding sides

= (4: 9)2

= 16 : 81

Thus option (D) 16: 81 is the correct answer.

☛ Check: NCERT Solutions for Class 10 Maths Chapter 6

Video Solution:

Sides of two similar triangles are in the ratio 4:9. Areas of these triangles are in the ratio (A) 2 : 3 (B) 4: 9 (C) 81: 16 (D) 16: 81

NCERT Class 10 Maths Solutions Chapter 6 Exercise 6.4 Question 9

Summary:

The sides of two similar triangles are in the ratio 4:9. Areas of these triangles are in the ratio 16: 81.

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Two triangles are Similar if the only difference is size (and possibly the need to turn or flip one around).

These triangles are all similar:

(Equal angles have been marked with the same number of arcs)

Some of them have different sizes and some of them have been turned or flipped.

For similar triangles:

All corresponding angles are equal

and

All corresponding sides have the same ratio

Also notice that the corresponding sides face the corresponding angles. For example the sides that face the angles with two arcs are corresponding.

Corresponding Sides

In similar triangles, corresponding sides are always in the same ratio.

For example:

Triangles R and S are similar. The equal angles are marked with the same numbers of arcs.

What are the corresponding lengths?

The lengths 7 and a are corresponding (they face the angle marked with one arc)
The lengths 8 and 6.4 are corresponding (they face the angle marked with two arcs)
The lengths 6 and b are corresponding (they face the angle marked with three arcs)

Calculating the Lengths of Corresponding Sides

We can sometimes calculate lengths we don't know yet.

Step 1: Find the ratio of corresponding sides
Step 2: Use that ratio to find the unknown lengths

Step 1: Find the ratio

We know all the sides in Triangle R, and
We know the side 6.4 in Triangle S

The 6.4 faces the angle marked with two arcs as does the side of length 8 in triangle R.

So we can match 6.4 with 8, and so the ratio of sides in triangle S to triangle R is:

6.4 to 8

Now we know that the lengths of sides in triangle S are all 6.4/8 times the lengths of sides in triangle R.

Step 2: Use the ratio

a faces the angle with one arc as does the side of length 7 in triangle R.

a = (6.4/8) × 7 = 5.6

b faces the angle with three arcs as does the side of length 6 in triangle R.

b = (6.4/8) × 6 = 4.8

Done!

Did You Know?

Similar triangles can help you estimate distances.

Answer

Verified

And the ratio of the sides of these two similar triangles be AB: PQ which is 4:9. If two triangles are similar, then the ratio of the areas of both the triangles is equal to the ratio of the squares of their corresponding sides. This means, the ratio of areas of triangles ABC and PQR will be $ \dfrac{{Ar\left( {\vartriangle ABC} \right)}}{{Ar\left( {\vartriangle PQR} \right)}} = \dfrac{{A{B^2}}}{{P{Q^2}}} = {\left( {\dfrac{{AB}}{{PQ}}} \right)^2} $ We already know that $ \dfrac{{AB}}{{PQ}} $ is equal to $ \dfrac{4}{9} $ Therefore, $ \dfrac{{Ar\left( {\vartriangle ABC} \right)}}{{Ar\left( {\vartriangle PQR} \right)}} = {\left( {\dfrac{4}{9}} \right)^2} = \dfrac{{16}}{{81}} = 16:81 $

So, the correct answer is “16:81”.

Note: Do not confuse similar triangles with congruent triangles, because congruent triangles have similar three sides and similar three angles which also mean similar areas, whereas the areas may or may not be the same in similar triangles. Congruent triangles are always similar whereas similar triangles may not be congruent always.