Two similar triangles are \( \triangle ABC \) and \( \triangle LMN \). If the area of \( \triangle ABC = 25 \) cm², the area of \( \triangle LMN = 36 \) cm², and BC = 2.5 cm, then the measure of MN (in cm) is:
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To find the measure of MN, we can use the property of similar triangles where the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Given:
- Area of \( \triangle ABC = 25 \, \text{cm}^2 \)
- Area of \( \triangle LMN = 36 \, \text{cm}^2 \)
- BC = 2.5 \, \text{cm}
We need to find MN. Let the side MN be denoted as \( x \).
The ratio of the areas of the triangles is given by:
\[\frac{\text{Area of } \triangle ABC}{\text{Area of } \triangle LMN} = \frac{25}{36}\]
This is equal to the square of the ratio of their corresponding sides:
\[\left(\frac{BC}{MN}\right)^2 = \left(\frac{2.5}{x}\right)^2\]
Thus, we have:
\[\frac{25}{36} = \left(\frac{2.5}{x}\right)^2\]
By taking the square root on both sides, we obtain:
\[\frac{5}{6} = \frac{2.5}{x}\]
Solving for \( x \):
\[x = \frac{2.5 \times 6}{5} = \frac{15}{5} = 3 \, \text{cm}\]
Therefore, the measure of MN is 3 cm.
Important Questions from Triangles
Two poles, 15 m and 30 m in height, stand upright in a playground. If their feet are 36 m apart, then find the distance between their tops.
How many triangles are there in the given figure?
How many triangles are there in the given figure?
Find the area (in square cm) of a right angled triangle whose altitude is 7 cm less than its base and its hypotenuse is 17 cm.
In ΔABC, if ∠A = 70° and ∠B = 70°, find the measure of exterior angle A.
