In the given figure, ∠ D = ∠ E and AD/DB = AE/EC. Prove that △ ABC is isosceles.

MathsCBSE Class 10 · Basic · Board exam 2024, set 430/2/1 · 2 marks · Very short answer

CBSE Class 10 Mathematics Basic 2024: question 24(b)

Set in the CBSE Class 10 board exam of 2024, paper 430/2/1, section B, question 24(b), for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q24(b) · 2 marks · Very short answerTriangles

In the given figure, $∠ D = ∠ E$ and $\frac{A D}{D B} = \frac{A E}{E C}$ . Prove that $△ A B C$ is isosceles.

Figure in the paper: Triangle ABC with apex A at the top and base BC horizontal at the bottom. Point D lies on side AB and point E lies on side AC. Segment DE is drawn. Angles at D (between DA and DE) and at E (between ED and EA) are marked as equal. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 430/2/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

The given equal ratios on two sides tell you something about line DE. What theorem gives this?

Hint 2

Once DE is parallel to BC, compare the angles at D and E with the angles at B and C.

Hint 3

Equal base angles mean which two sides are equal?

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

In $△ A B C$ , $\frac{A D}{D B} = \frac{A E}{E C}$ (given), so $D E ∥ B C$ [converse of Basic Proportionality Theorem].
½ mark
Hence $∠ A D E = ∠ A B C$ and $∠ A E D = ∠ A C B$ [corresponding angles, $D E ∥ B C$ ].
½ mark
But $∠ A D E = ∠ A E D$ (given), so $∠ B = ∠ C$ .
½ mark
Therefore $A B = A C$ [sides opposite equal angles of a triangle are equal], so $△ A B C$ is isosceles.
½ mark

Answer: $A B = A C$ , so $△ A B C$ is isosceles

Where the marks are

Uses the converse of BPT to get $D E ∥ B C$ (½ mark)
States the corresponding angles $∠ D = ∠ B$ and $∠ E = ∠ C$ (½ mark)
Concludes $∠ B = ∠ C$ (½ mark)
Gets $A B = A C$ , so the triangle is isosceles (½ mark)

Revise the chapter

Chapter: Triangles (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Triangles, one at a time, marked against the answer key and marking points.Back to the full paper2024 board paper, set 430/2/1 (Basic): every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 10 Mathematics Basic 2024: question 24(b)

Hints

Step-by-step solution

Revise the chapter

More questions from Triangles