Triangle ABC is an isosceles right triangle right angled at B.DE is a line parallel to BC which intersects AB in D and AC in E.What ratio are the sides of triangle ADE in?a)3:4:5b)1:1:1c)1:1:√2d)1:√3:2Correct answer is option 'C'. Can you explain this answer?

Question

Triangle ABC is an isosceles right triangle right angled at B.DE is a line parallel to BC which intersects AB in D and AC in E.What ratio are the sides of triangle ADE in?a)3:4:5b)1:1:1c)1:1:&#8730;2d)1:&#8730;3:2Correct answer is option 'C'. Can you explain this answer?

Accepted Answer

Since AB =BC and the triangles are similar, AB/AD=BC/DE
Gives AD=DE
So by Pythagoras theorem,
H² = P² + B²
H²= 2x²
H = √2x
So the sides are in the ratio x:x: √2x which is 1:1: √2

Answer

Two sides of a triangle are given equal let them=a so the third side (hypotenuse)=square root of (a square +a square=square root 2.a so the sides of a isosceles right Triangle are in ratio 1:1: root 2. the intersection of sides is not related with the question asked .

Answer

Explanation:

Given:
- Triangle ABC is an isosceles right triangle with angle B as 90 degrees.
- DE is a line parallel to BC, intersecting AB at D and AC at E.

Properties of Isosceles Right Triangle:
- In an isosceles right triangle, the two equal sides are the legs, and the hypotenuse is √2 times the length of each leg.
- In triangle ABC, AB = AC as it is an isosceles triangle.
- Therefore, the ratio of sides AB:AC is 1:1.

Using Similarity of Triangles:
- Since DE is parallel to BC, triangle ADE is similar to triangle ABC by AA similarity.
- The corresponding sides of similar triangles are proportional.
- Therefore, the ratio of sides AD:AB and AE:AC is equal.

Ratio of Sides in Triangle ADE:
- Since AB = AC, the ratio of AD:AB:AE:AC is 1:1:1:1.
- However, we need to find the ratio of sides in triangle ADE, so we consider only AD, AE, and DE.
- As AD = AE and DE is common, the ratio of sides in triangle ADE is 1:1:2.
Therefore, the ratio of sides in triangle ADE is 1:1:2, which corresponds to option 'C'.

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