Let ABCD be a quadrilateral. If E and F are the mid points of the diagonals AC and BD respectively and $ (\vec{AB}-\vec{BC})+(\vec{AD}-\vec{DC})=k \vec{FE} $, then k is equal to
Let ABCD be a quadrilateral. If E and F are the mid points of the diagonals AC and BD respectively and $ (\vec{AB}-\vec{BC})+(\vec{AD}-\vec{DC})=k \vec{FE} $, then k is equal to
- -4
- -2
- 2
- 4
The Correct Option is A
Solution and Explanation
Top Questions on Quadrilaterals
Complete the following activity to prove that the sum of squares of diagonals of a rhombus is equal to the sum of the squares of the sides.
Given: PQRS is a rhombus. Diagonals PR and SQ intersect each other at point T.
To prove: PS\(^2\) + SR\(^2\) + QR\(^2\) + PQ\(^2\) = PR\(^2\) + QS\(^2\)
Activity: Diagonals of a rhombus bisect each other.
In \(\triangle\)PQS, PT is the median and in \(\triangle\)QRS, RT is the median.
\(\therefore\) by Apollonius theorem,
\[\begin{aligned} PQ^2 + PS^2 &= \boxed{\phantom{X}} + 2QT^2 \quad \dots \text{(I)} \\ QR^2 + SR^2 &= \boxed{\phantom{X}} + 2QT^2 \quad \dots \text{(II)} \\ \text{Adding (I) and (II),} \quad PQ^2 + PS^2 + QR^2 + SR^2 &= 2(PT^2 + \boxed{\phantom{X}}) + 4QT^2 \\ &= 2(PT^2 + \boxed{\phantom{X}}) + 4QT^2 \quad (\text{RT = PT}) \\ &= 4PT^2 + 4QT^2 \\ &= (\boxed{\phantom{X}})^2 + (2QT)^2 \\ \therefore \quad PQ^2 + PS^2 + QR^2 + SR^2 &= PR^2 + \boxed{\phantom{X}} \\ \end{aligned}\]
- Maharashtra Class X Board - 2025
- Mathematics
- Quadrilaterals
- Each angle of a rectangle is
- UP Board X - 2025
- Chitrakala
- Quadrilaterals
- If \( A(3, 1, -1) \), \( B\left(\frac{5}{3}, \frac{7}{3}, \frac{1}{3}\right) \), \( C(2, 2, 1) \), and \( D\left(\frac{10}{3}, \frac{2}{3}, \frac{-1}{3}\right) \) are the vertices of a quadrilateral ABCD, then its area is:
- JEE Main - 2024
- Mathematics
- Quadrilaterals
- A quadrilateral \(ABCD\) is inscribed in a circle such that \(AB :CD\) = \(2:1\) and \(BC:AD = 5: 4\). If \(AC\) and \(BD\) intersect at the point \(E\),then \(AE:CE\) equals
- CAT - 2023
- Quantitative Aptitude
- Quadrilaterals
- A quadrilateral ABCD has diagonal AC of length 28 cm. If the perpendiculars drawn from vertices B and D to AC are 30 cm and 21 cm respectively, then find its area.
- CUET (PG) - 2023
- Mathematics
- Quadrilaterals
Questions Asked in JEE Main exam
The most stable carbocation from the following is:
- JEE Main - 2025
- Organic Chemistry
Consider the following sequence of reactions :
Molar mass of the product formed (A) is ______ g mol\(^{-1}\).- JEE Main - 2025
- Organic Chemistry
- The distance of the line \( \frac{x - 2}{2} = \frac{y - 6}{3} = \frac{z - 3}{4} \) from the point \( (1, 4, 0) \) along the line \( \frac{x}{1} = \frac{y - 2}{2} = \frac{z + 3}{3} \) is:
- JEE Main - 2025
- Geometry and Vectors
Let one focus of the hyperbola \( H : \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 \) be at \( (\sqrt{10}, 0) \) and the corresponding directrix be \( x = \dfrac{9}{\sqrt{10}} \). If \( e \) and \( l \) respectively are the eccentricity and the length of the latus rectum of \( H \), then \( 9 \left(e^2 + l \right) \) is equal to:
- JEE Main - 2025
- Conic sections
- A tiny metallic rectangular sheet has length and breadth of 5 mm and 2.5 mm, respectively. Using a specially designed screw gauge which has pitch of 0.75 mm and 15 divisions in the circular scale, you are asked to find the area of the sheet. In this measurement, the maximum fractional error will be \( \frac{x}{100} \), where \( x \) is:
- JEE Main - 2025
- Units and measurement