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

CBSE Class 10 Mathematics Basic 2024: question 21

Set in the CBSE Class 10 board exam of 2024, paper 430/1/1, section B, question 21, for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q21 · 2 marks · Very short answerTriangles

D is a point on the side BC of $Δ$ ABC such that $∠$ ADC = $∠$ BAC. Show that $A C^{2} = B C \times D C$ .

Figure in the paper: Triangle ABC with vertex A at top, base BC horizontal. Point D lies on BC, and segment AD is drawn from A to D. See it in the official paper.

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

Hints

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

Hint 1

Look for two triangles that both contain the side AC.

Hint 2

Compare $△ A B C$ and $△ D A C$ : one pair of angles is given equal; which other angle do they share?

Hint 3

Once they are similar, match sides opposite equal angles and write the ratio involving AC twice.

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 $△ A B C$ and $△ D A C$ : $∠ B A C = ∠ A D C$ [given]
½ mark
$∠ A C B = ∠ D C A$ [common angle at C]
½ mark
$∴ △ A B C \sim △ D A C$ [AA similarity criterion]
½ mark
Corresponding sides of similar triangles are proportional, so $\frac{A C}{D C} = \frac{B C}{A C}$ . Cross-multiplying gives $A C^{2} = B C \times D C$ . Hence proved.
½ mark

Answer: $△ A B C \sim △ D A C$ (AA), so $A C^{2} = B C \times D C$

Where the marks are

States $∠ B A C = ∠ A D C$ (given) (½ mark)
States $∠ C$ is common (½ mark)
Concludes $△ A B C \sim △ D A C$ by AA (½ mark)
Writes the ratio $\frac{A C}{D C} = \frac{B C}{A C}$ and obtains $A C^{2} = B C \times D C$ (½ mark)

Revise the chapter

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

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 430/1/1 (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 21

Hints

Step-by-step solution

Revise the chapter

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