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प्रश्न
Construct the circumcircle and incircle of an equilateral ∆XYZ with side 6.5 cm and centre O. Find the ratio of the radii of incircle and circumcircle.
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उत्तर
Steps of Construction (to draw the circumcircle):
1. Draw an equilateral triangle ABC with each side 6.5 cm.
2. Draw the perpendicular bisectors of AB and BC. Let these meet at the point O.
3. With O as centre and OB as radius, draw a circle. This circle is the circumcircle of triangle ABC.
Steps of Construction (to draw the incircle):
1. Draw the angle bisectors of \[\angle\]CAB. It passes through the point O.
2. From point O, draw a perpendicular on AB. Let this meet AB in D.
3. With O as centre and OD as radius, draw a circle. This circle is the incircle of triangle ABC.
\[\frac{Radius\ of\ incircle}{Radius\ of\ circumcircle} = \frac{2}{4} = \frac{1}{2}\]
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Draw a line segment of length 7.6 cm and divide it in the ratio 5:8. Measure the two parts. Give the justification of the construction.
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Construct an isosceles triangle whose base is 8 cm and altitude 4 cm and then another triangle whose side are `1 1/2` times the corresponding sides of the isosceles triangle.
Give the justification of the construction
Divide a line segment of length 9 cm internally in the ratio 4 : 3. Also, give justification of the construction.
Draw a right triangle in which the sides (other than hypotenuse) are of lengths 5 cm and 4 cm. Then construct another triangle whose sides are 5/3th times the corresponding sides of the given triangle.
Construct an isosceles triangle whose base is 8 cm and altitude 4 cm and then another triangle whose sides are 3/2 times the corresponding sides of the isosceles triangle.
Construct a triangle similar to a given ΔXYZ with its sides equal to (3/4)th of the corresponding sides of ΔXYZ. Write the steps of construction.
Draw a line segment AB of length 7 cm. Using ruler and compasses, find a point P on AB such that `(AP)/(AB) = 3/5`.
Draw a line segment of length 7.6 cm and divide it in the ratio 5 : 8. Measure the two parts.
Draw a ∆ABC in which AB = 4 cm, BC = 5 cm and AC = 6 cm. Then construct another triangle whose sides are\[\frac{3}{5}\] of the corresponding sides of ∆ABC ?
∆PQR ~ ∆LTR. In ∆PQR, PQ = 4.2 cm, QR = 5.4 cm, PR = 4.8 cm. Construct ∆PQR and ∆LTR, such that `"PQ"/"LT" = 3/4`.
Find the co-ordinates of the points of trisection of the line segment AB with A(2, 7) and B(–4, –8).
Given A(4, –3), B(8, 5). Find the coordinates of the point that divides segment AB in the ratio 3 : 1.
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If the point P(6, 7) divides the segment joining A(8, 9) and B(1, 2) in some ratio, find that ratio.
Solution:
Point P divides segment AB in the ratio m : n.
A(8, 9) = (x1, y1), B(1, 2) = (x2, y2) and P(6, 7) = (x, y)
Using Section formula of internal division,
∴ `7 = (m(square) - n(9))/(m + n)`
∴ 7m + 7n = `square` + 9n
∴ 7m – `square` = 9n – `square`
∴ `square` = 2n
∴ `m/n = square`
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