✓ Board-verified↻ asked 2×
Mathematics · 2023 · 3 marks
CBSE 2023 · Region 5 · Set 1 · Q28
In the given figure, CD is the perpendicular bisector of AB. EF is perpendicular to CD. AE intersects CD at G. Prove that $\displaystyle \frac{C F}{C D}=\frac{F G}{D G}$.
In the given figure, ABCD is a parallelogram. BE bisects CD at M and intersects AC at L. Prove that EL = 2BL.
In the given figure, CD is the perpendicular bisector of AB. EF is perpendicular to CD. AE intersects CD at G. Prove that $\displaystyle \frac{C F}{C D}=\frac{F G}{D G}$.
In the given figure, ABCD is a parallelogram. BE bisects CD at M and intersects AC at L. Prove that EL = 2BL.
Marking-scheme solution
\(\displaystyle \boldsymbol{\Delta}\) EFG \(\displaystyle \boldsymbol{\sim} \boldsymbol{\Delta} \mathbf{~ A D G}\)
\[\Rightarrow \frac{E F}{A D}=\frac{F G}{D G} \longrightarrow \text { (i) }
\]
\(\displaystyle \boldsymbol{\Delta}\) EFC \(\displaystyle \boldsymbol{\sim} \boldsymbol{\Delta}\) BDC
\[\Rightarrow \frac{E F}{B D}=\frac{C F}{C D}
\]
\(\displaystyle \Rightarrow \frac{E F}{A D}=\frac{C F}{C D} \quad\{\mathrm{BD}=\mathrm{AD}\}\) \(\displaystyle \_\_\_\_\) (ii)
Using (i) and (ii)
\[\frac{F G}{D G}=\frac{C F}{C D}
\]
\(\displaystyle \boldsymbol{\Delta}\) ALE ~ \(\displaystyle \boldsymbol{\Delta}\) CLB
\[\Rightarrow \frac{A L}{C L}=\frac{E L}{B L}
\]
(i)
Also \(\displaystyle \boldsymbol{\Delta} \mathbf{C L M} \boldsymbol{\sim} \boldsymbol{\Delta} \mathbf{~ A L B}\)
\[\Rightarrow \frac{A L}{C L}=\frac{A B}{C M}
\]
\(\displaystyle \Rightarrow \frac{\boldsymbol{A} \boldsymbol{L}}{\boldsymbol{C} \boldsymbol{L}}=\frac{\boldsymbol{C} \boldsymbol{D}}{\boldsymbol{C} \boldsymbol{M}}\) \(\displaystyle \_\_\_\_\)
\(\displaystyle \_\_\_\_\) (ii)
Using (i) and (ii)
\[\frac{E L}{B L}=\frac{2 C M}{C M}
\]
⇒ EL = 2BL
More from Coordinate Geometry
- The line segment joining the points P(-4,-2) and Q(10,4) is divided by y-axis in the ratio2026 · asked 3×
- Find the ratio in which the y-axis divides the line segment joining the points (5, -6) and (-1, -4). Also…2025 · asked 3×
- If Q(0,1) is equidistant from P(5,-3) and R(x, 6), find the values of x.2023 · asked 3×
- The coordinates of the end points of the line segment AB are A(-2, -2) and B(2,-4). P is the point on AB such…2025 · asked 3×
- Find a relation between x and y such that P(x, y) is equidistant from the points A(3, 5) and B(7, 1). Hence,…2025 · asked 3×
- If the diagonals of a quadrilateral divide each other proportionally, then it is a:2024 · asked 3×
- Two of the vertices of Δ PQR are P(-1,5) and Q(5,2). The coordinates of a point which divides PQ in the ratio…2025 · asked 3×
- A garden is in the shape of a square. The gardener grew saplings of Ashoka tree on the boundary of the garden…2024 · asked 3×
CBSE Class 10 Mathematics past-paper question from the 2023board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.