Question
P is a point equidistant from two lines l and m intersecting at a point A (see figure). Show that $AP$ bisects the angle between them.
✓
Answer
Given: $P$ is a point equidistant from two lines $I$ and $m$ intersect at a point $A$.
To Prove: AP bisects the angle between them. i.e. $\angle \mathrm{PAB}=\angle \mathrm{PAC}$
Proof: Let $P B$ and $P C$ be perpendiculars from $P$ on lines $I$ and $m$ respectively. Since $P$ is equidistant from lines $I$ and $m$.
Therefore, $P B=P C$
In $\triangle P A B$ and $\triangle P A C$, we have
$\mathrm{PB}=\mathrm{PC} \text { [Given] }$
$\angle \mathrm{PBA}=\angle \mathrm{PCA}\left[\text { Each equal to } 90^{\circ}\right. \text { ] }$
$\text { and, } \mathrm{PA}=\mathrm{PA}[\mathrm{Common}]$
$\triangle \mathrm{PAB} \cong \triangle \mathrm{PAC}[\mathrm{By} \mathrm{~RHS} \text { congruence criterion] }$
$\Rightarrow \angle \mathrm{PAB}=\angle \mathrm{PAC}$
Hence Proved
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Evaluate: $(995)^2$
→Expand:
$(3 x+2)^3$
→$(3 x+2)^3$
Rationalise the denominator of $\frac{5}{\sqrt{3}-\sqrt{5}}$
→A cuboidal water tank is $6 \ m$ long, $5 \ m$ wide and $4.5 \ m$ deep. How many litres of water can it hold? (Given, $1 \mathrm{~m}^3=1000$ litres.)
→The linear equation that converts Fahrenheit $(F)$ to Celsius $(C)$ is given by the relation, $\text{c}=\frac{5\text{F}-160}{9}$ If the temperature is $35^\circ C$, what is the temperature in Fahrenheit?
→Find the curved surface area of a cone, if its slant height is $60\ cm$ and the radius of its base is $21\ cm.$
→Factorise:
$x^3-3 x^2+3 x+7$
→$x^3-3 x^2+3 x+7$
Simplify: $\Big(\frac{32}{243}\Big)^{-\frac{4}{5}}$
→Factorise : $8 x^3+y^3+27 z^3-18 x y z$
→Write the following in the expanded form: $\Big(\frac{\text{x}}{\text{y}}+\frac{\text{y}}{\text{z}}+\frac{\text{z}}{\text{x}}\Big)^2$
→