In a parallelogram with vertices Aa-Bb-Cc-Dd-Eeis midpoint of digonal AC then

The correct options are A e=a+c2 C e=d+b2 Given #160;E(e) is the mid point of diagonal ACby parallelogram law #160;diagonal AC AC=a⃗+c⃗ posit ...

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Byju's Answer

Standard IX

Mathematics

The Mid-Point Theorem

In a parallel...

Question

In a parallelogram with vertices A(\overline a),B(\overline b),C(\overline c),D(\overline d) $,,$ E(\overline e)$ is midpoint of digonal AC then

¯ ¯ ¯ e = \frac{¯ ¯ ¯ a + ¯ ¯ c}{2}

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¯ ¯ ¯ e = \frac{¯ ¯ ¯ d + ¯ ¯ b}{2}

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¯ ¯ ¯ e = \frac{¯ ¯ ¯ a - ¯ ¯ c}{2}

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¯ ¯ ¯ e = \frac{¯ ¯ ¯ d - ¯ ¯ b}{2}

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Solution

The correct options are
A $¯ ¯ ¯ e = \frac{¯ ¯ ¯ a + ¯ ¯ c}{2}$
C $¯ ¯ ¯ e = \frac{¯ ¯ ¯ d + ¯ ¯ b}{2}$
Given
E(e) is the mid point of diagonal AC
by parallelogram law
diagonal AC
$A C = \to a + \to c$
position vector of E
$\to e = \frac{\to a + \to c}{2}$
Hence option A is correct

diagonal AC in other triangle ACD
$A C = \to d + \to b$
position vector of E
$\to e = \frac{\to d + \to b}{2}$

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Q. If x² − 1 is a factor of ax⁴ + bx³ + cx² + dx + e, then

(a) a + c + e = b + d

(b) a + b +e = c + d

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Q. ABCD is a parallelogram. E is a point on BA such that BE = 2 EA and F is a point on DC such that DF = 2 FC. Prove that AE CF is a parallelogram whose area is one third of the area of parallelogram AB CD.

If A = {3, 6, 9, 12, 15, 18, 21}

B = {2, 4, 6, 8, 10, 12, 14, 16}

C = {5, 10, 15, 20}

$\begin{matrix} A & B 1. B - C & A . {5, 10, 20} 2. A - C & B . {3, 9, 15, 21} 3. C - (A - B) & C . {3, 6, 9, 12, 18, 21} 4. C \cap (A - B) & D . {15} E . {2, 4, 6, 8, 12, 14, 16} \end{matrix}$

Q. Show that

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} a & b & c & d & e & f f & a & b & c & d & e e & f & a & b & c & d d & e & f & a & b & c c & d & e & f & a & b b & c & d & e & f & a \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ \begin{matrix} A & B & C C & A & B B & C & A \end{matrix} ∣ ∣ ∣ ∣

where

A = a^{2} - d^{2} + 2 c e - 2 b f

B = e^{2} - b^{2} + 2 a c - 2 d f

C = c^{2} - f^{2} + 2 a e - 2 b d

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In a parallelogram with vertices A(\overline a),B(\overline b),C(\overline c),D(\overline d),,E(\overline e)$ is midpoint of digonal AC then

In a parallelogram with vertices A(\overline a),B(\overline b),C(\overline c),D(\overline d) $,,$ E(\overline e)$ is midpoint of digonal AC then