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This is the second of our many quizzes on Solving for a side in a right triangle. Click on this to access more such competitive quizzes on Geometry.

Solving for a side in a right triangle Quiz 2

mathematics

 A medic team sees an injured man at a horizontal distance of 4 Km from where their helicopter is. If they are flying at a height of 3 km from the ground. What is the shortest distance to shoot their helping supplies to the man?

 Consider the scenario like a right angled triangle and use Pythagoras theorem, where ${{(Hypotenuse)}^{2}}={{(Base)}^{2}}+{{(Perpendicular)}^{2}}$

  Here a right angled triangle is formed, such that the horizontal distance between the man and point above which the helicopter is flying and the vertical distance of the helicopter can be the sides of the said triangle as shown. $ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/bed62c7b-ff65-425d-aa36-bb3d8b91f2a28191638258750501015.png"/> $ \\ $ The direct length between the helicopter and man can be found using Pythagoras theorem $ \\ $ According to Pythagoras theorem, $ \\ {{(\text{Hypotenuse})}^{2}}={{(\text{Base})}^{2}}+{{(\text{Perpendicular})}^{2}} \\ $  Replacing the values $ \\ {{(\text{Hypotenuse})}^{2}}={{(4)}^{2}}+{{(3)}^{2}} \\ {{(\text{Hypotenuse})}^{2}}=16+9=25 \\ \text{Hypotenuse}=\sqrt{25}=5km \\ $ 5km is the shortest distance.

 A civil engineer has a contract whereby he has to build a bridge with a length of 60 m. If the angle of elevation is ${{30}^{\circ }}$ what will be the maximum height of the bridge?

 Since the angle of elevation and a hypotenuse of the right angle triangle is given, the value of perpendicular can be found using $\operatorname{Sin}\theta =\dfrac{P}{H}$.

  It is known that the angle of elevation of the bridge is ${{30}^{\circ }}$ $ \\ $ The figure will be as $ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/1291b517-f7e6-4c1e-9808-8cbb5f9225134837530349844619598.png"/> $ \\ $ We need to find the length of AB  $ \\ \operatorname{Sin}\theta =\dfrac{P}{H} \\ $  Here $\sin (30)=\dfrac{AB}{60}$ $ \\  \Rightarrow \dfrac{1}{2}=\dfrac{AB}{60} \\ \Rightarrow AB=\dfrac{60}{2}=30m $

 A triangle is made by enclosing a plane area by three squares such that the length of two squares is 6 cm and 9 cm. What would be the length of the largest side? 

The squares are such places that they form a figure as shown $ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/ac0a5a94-9c1a-464c-a744-84c76e48afd11648373985650686635.png"/> $ \\ $ Since the figure makes a right angled triangle hence, $ \\ $ According to Pythagoras theorem, $ \\ {{(Hypotenuse)}^{2}}={{(Base)}^{2}}+{{(Perpendicular)}^{2}} \\ $  Replacing the values $ \\ {{(Hypotenuse)}^{2}}={{(6)}^{2}}+{{(9)}^{2}} \\ {{(Hypotenuse)}^{2}}=36+81=117 \\ Hypotenuse=\sqrt{117}cm \\ $ Length of the largest side is $\sqrt{117}cm$.

 A mountaineer ties a rope across two poles to practice, such that one side is lower than the other. If one side is 20 m and the other pole is 12 m. The distance between the mounts of two poles is 40 m. Find the length of pole required.

 <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/663b3d38-08a0-486b-a3e7-af1f000fb44d4459768338791544689.png"/> $ \\ $ Here poles are denoted by AD and BC and the distance between the poles is DC. $ \\ AE=AD-BC \\ AE=20-12 \\ AE=8 \\ $  Since $\vartriangle AEB$ is right angled $ \\ $ According to Pythagoras theorem, $ \\ {{(\text{Hypotenuse})}^{2}}={{(Base)}^{2}}+{{(Perpendicular)}^{2}} \\ $  Replacing the values, ${{(\text{Hypotenuse})}^{2}}={{(8)}^{2}}+{{(40)}^{2}} \\ {{(\text{Hypotenuse})}^{2}}=64+1600=1664 \\ \text{Hypotenuse}=\sqrt{1664} = \sqrt{2^7\cdot 13} \\ \Rightarrow \sqrt{2^6}\sqrt{2\cdot 13} \\ \Rightarrow 8\sqrt{26} \\ $ The length of the pole is $8\sqrt{26}m$.

 Steps are constructed such that the depth of the individual step is 6 cm and the base is 8 cm. If a length of 200 cm is to be converted to steps, find the number of steps needed.

 Consider the scenario like a right angled triangle and use Pythagoras theorem, where ${{(Hypotenuse)}^{2}}={{(Base)}^{2}}+{{(Perpendicular)}^{2}}$ 

 <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/8008e7ec-2e23-4474-9792-72f41b4bc5b8719041929077385784.png"/> $ \\ $ According to Pythagoras theorem, $ \\ {{(Hypotenuse)}^{2}}={{(Base)}^{2}}+{{(Perpendicular)}^{2}} \\ $ Replacing the values $\\ {{(Hypotenuse)}^{2}}={{(6)}^{2}}+{{(8)}^{2}} \\ {{(Hypotenuse)}^{2}}=36+64=100 \\ Hypotenuse=\sqrt{100}=10cm \\ $ If one step is of 10 cm, for 200 cm $ \\ \text{Number of steps= }\dfrac{\text{200}}{10}=20$