Question : In $\triangle$ABC, D is the median from A to BC. AB = 6 cm, AC = 8 cm, and BC = 10 cm.The length of median AD (in cm) is:
Option 1: 4.5
Option 2: 5
Option 3: 4
Option 4: 3
Correct Answer: 5
Solution : AB = 6 cm BC = 10 cm AC = 8 cm AD is the median bisects BC Pythagoras theorem, AB$^2$ + AC$^2$ = BC$^2$ Circumcentre theorem Circumcentre of a right-angled triangle lies on the mid-point of the hypotenuse AD = BD = DC AB$^2$ + AC$^2$ = BC$^2$ $6^2 + 8^2 = 10^2$ Since it obeys the Pythagoras theorem It is a right-angled triangle Circumcentre of a right-angled triangle lies on the mid-point of the hypotenuse AD = BD = DC AD is a median which bisects BC into two equal parts BD = DC = 5 cm AD = BD = DC = 5 cm. $\therefore$ The length of the median AD is 5 cm. Hence, the correct answer is 5 cm.
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Question : Let $\triangle ABC \sim \triangle RPQ$ and $\frac{{area}(\triangle {ABC})}{{area}(\triangle {PQR})}=\frac{4}{9}$. If AB = 3 cm, BC = 4 cm and AC = 5 cm, then RP (in cm) is equal to:
Question : The sides AB, BC, and AC of a $\triangle {ABC}$ are 12 cm, 8 cm, and 10 cm respectively. A circle is inscribed in the triangle touching AB, BC, and AC at D, E, and F respectively. The difference between the lengths of AD and CE is:
Question : Let $\triangle {ABC} \sim \triangle {RPQ}$ and $\frac{\operatorname{ar}(\triangle A B C)}{\operatorname{ar}(\triangle P Q R)}=\frac{4}{9}$. If ${AB}=3 {~cm}, {BC}=4 {~cm}$ and ${AC}=5 {~cm}$, then ${PQ}$ (in ${cm}$ ) is equal to:
Question : In $\triangle$ ABC, $\angle$ C = 90$^{\circ}$. M and N are the midpoints of sides AB and AC, respectively. CM and BN intersect each other at D and $\angle$ BDC = 90$^{\circ}$. If BC = 8 cm, then the length of BN is:
Question : In $\triangle \mathrm{ABC}$, AB = AC, and D is a point on side AC such that BD = BC. If AB = 12.5 cm and BC = 5 cm, then what is the measure of DC?
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