Geometry - Study Mode

[#211] In the following figure, AD bisects angle BAC. Find the length (in cm) of BD.
Correct Answer

(A) 4

Explanation

Solution: $$eqalign{
& AD{ ext{ is angle bisector of }}angle A cr
& herefore frac{{AB}}{{AC}} = frac{{BD}}{{DC}} cr
& Rightarrow frac{6}{{2x - 3}} = frac{{x - 2}}{x} cr
& Rightarrow 6x = 2{x^2} - 4x - 3x + 6 cr
& Rightarrow 2{x^2} - 13x + 6 = 0 cr
& Rightarrow 2{x^2} - 12x - x + 6 = 0 cr
& Rightarrow 2xleft( {x - 6}
ight) - 1left( {x - 6}
ight) = 0 cr
& Rightarrow left( {x - 6}
ight)left( {2x - 1}
ight) = 0 cr
& x - 6 = 0 cr
& x = 6 cr
& 2x - 1 = 0 cr
& x = frac{1}{2}left( {{ ext{not valied}}}
ight) cr
& herefore BD = x - 2 = 6 - 2 = 4 cr} $$

[#212] In the given figure, if $$frac{{{ ext{QR}}}}{{{ ext{XY}}}} = frac{{14}}{9}$$ xa0 and PY = 18 cm, then what is the value (in cm) of PQ?
Correct Answer

(A) 28

Explanation

Solution: $$eqalign{
& angle QXY = {120^ circ } cr
& angle PXY = {60^ circ } cr
& Delta PXY sim Delta PRQ cr
& herefore frac{{PY}}{{PQ}} = frac{{XY}}{{QR}} cr
& frac{{18}}{{PQ}} = frac{9}{{14}} cr
& PQ = frac{{18 imes 14}}{9} = 28,{ ext{cm}} cr} $$

[#213] Number of circles that can be drawn through three non-collinear points are
Correct Answer

(A) exactly one

Explanation

Solution: Only one (1) circle can be drawn through three non-collinear points.

[#214] The diagonal of a quadrilateral shaped field is 24 m and the perpendiculars dropped on it from the remaining opposite vertices are 8 m and 13 m. The area of the field is?
Correct Answer

(A) 252 m 2

Explanation

Solution: Let ABCD is quadrilateral and its BD diagonal BD = 24 metres And, AM = 8 metres CN = 13 metres $$eqalign{
& { ext{Area of }}square ABCD = { ext{ar}}left( {Delta ABD}
ight) + { ext{ar}}left( {Delta BCD}
ight) cr
& = left( {frac{1}{2} imes BD imes AM}
ight) + left( {frac{1}{2} imes BD imes CN}
ight) cr
& = frac{1}{2} imes BDleft[ {AM + CN}
ight] cr
& = frac{1}{2} imes 24left[ {8 + 13}
ight] cr
& = 12 imes 21 cr
& { ext{Area of }}square ABCD = 252{ ext{ metr}}{{ ext{e}}^2} cr} $$

[#215] The exterior angle of a triangle is 115° and the corresponding interior opposite angles are in the ratio 2 : 3. The measure of greatest angle of the triangle is:
Correct Answer

(C) 69°

Explanation

Solution: 5x = 115° (by exterior angle theorem) x = 23° 2x = 46° 3x = 69° ∠ABC = 65° The greatest angle = 69°