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221.

Prove that an angle inscribed in a semi-circle is a right angle.

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222. Using vectors, prove that if two medians of a triangle ABC be equal, then it is an isosceles triangle.

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223. Using vectors, prove that the perpendicular bisectors of the sides of a triangle are concurrent.

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224. Prove that the median to the base of isosceles triangle is perpendicular to the base.

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225. If the median of the base of a triangle is perpendicular on the base, then prove that the triangle is an isosceles.

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226. (i) Prove that cos (α + β) )= cos α cos β – sin α sin β.
(ii) Prove that cos (α – p) = cos α cos β + sin α sin β.

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Let ABCD be the given rectangle.
Take A as origin.

Let

AB with rightwards arrow on top space equals space straight b with rightwards arrow on top comma space space space AD with rightwards arrow on top space equals space straight d.

Now,

AC with rightwards arrow on top space equals space AB with rightwards arrow on top space plus space BC with rightwards arrow on top space equals space AB with rightwards arrow on top space plus space AD with rightwards arrow on top space equals space straight b with rightwards arrow on top space plus space straight d with rightwards arrow on top

Also,

BD with rightwards arrow on top space equals space straight P. straight V. space of space straight D space minus space straight P. straight V. space of space straight B space equals space straight d with rightwards arrow on top space minus space straight b with rightwards arrow on top

.
Now diagonals AC and BD are perpendicular if

AC with rightwards arrow on top

is perpendicular to

BD with rightwards arrow on top.

i.e. iff

AC with rightwards arrow on top. space BD with rightwards arrow on top space equals space 0

i.e. iff

left parenthesis straight b with rightwards arrow on top plus straight d with rightwards arrow on top right parenthesis. space left parenthesis straight d with rightwards arrow on top space minus straight b with rightwards arrow on top right parenthesis space equals space 0

i.e. iff

left parenthesis straight d with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis. space left parenthesis straight d with rightwards arrow on top space minus space straight b with rightwards arrow on top right parenthesis space equals space 0

i.e. iff

straight d with rightwards arrow on top squared space minus space straight b with rightwards arrow on top squared space equals space 0

i.e. iff

open vertical bar straight d with rightwards arrow on top close vertical bar squared space equals space open vertical bar straight b with rightwards arrow on top close vertical bar squared

i.e. iff

AD squared space equals space AB squared

i.e. iff AD = AB
i.e. iff rectangle ABCD is a square.
Hence the result.

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228. Using vectors, prove that a parallelogram whose diagonals are equal is a rectangle.

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229. Prove using vectors: The quadrilateral obtained by joining mid-points of adjacent sides of a rectangle is a rhombus.

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230. Show that the angle between two diagonals of a cube is

cos to the power of negative 1 end exponent space open parentheses 1 third close parentheses.

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