Finding a shorter side questions with answers

Questions

1. Question 1Whole-number answers

   In the diagram, $\angle KLM = 90^\circ$. Find $x$.

   Show the worked solution

   1. Step 1: Name the hypotenuse: the side opposite the right angleThe hypotenuse is $KM$, opposite the right angle at $L$.
   2. Step 2: Use Pythagoras' theorem$KM^2 = KL^2 + LM^2$$x^2 = 70^2 - 42^2$$= 4900 - 1764$$= 3136$
   3. Step 3: Square-root only at the end; it must be shorter than the hypotenuse$x = \sqrt{3136}$$= 56$ mCheck: $56 < 70$, shorter than the hypotenuse.
2. Question 2Whole-number answers

   In the diagram, $\angle PQR = 90^\circ$. Find $x$.

   Show the worked solution

   1. Step 1: Name the hypotenuse: the side opposite the right angleThe hypotenuse is $PR$, opposite the right angle at $Q$.
   2. Step 2: Use Pythagoras' theorem$PR^2 = PQ^2 + QR^2$$x^2 = 74^2 - 24^2$$= 5476 - 576$$= 4900$
   3. Step 3: Square-root only at the end; it must be shorter than the hypotenuse$x = \sqrt{4900}$$= 70$ mCheck: $70 < 74$, shorter than the hypotenuse.
3. Question 3Answers to 3 significant figures

   In the diagram, $\angle ABC = 90^\circ$. Find $x$.

   Show the worked solution

   1. Step 1: Name the hypotenuse: the side opposite the right angleThe hypotenuse is $AC$, opposite the right angle at $B$.
   2. Step 2: Use Pythagoras' theorem$AC^2 = AB^2 + BC^2$$x^2 = 6^2 - 2^2$$= 36 - 4$$= 32$
   3. Step 3: Square-root only at the end; it must be shorter than the hypotenuse$x = \sqrt{32}$$= 5.6568\ldots$$= 5.66$ cm (3 s.f.)Check: $5.66 < 6$, shorter than the hypotenuse.
4. Question 4Answers to 3 significant figures

   In the diagram, $\angle PQR = 90^\circ$. Find $x$.

   Show the worked solution

   1. Step 1: Name the hypotenuse: the side opposite the right angleThe hypotenuse is $PR$, opposite the right angle at $Q$.
   2. Step 2: Use Pythagoras' theorem$PR^2 = PQ^2 + QR^2$$x^2 = 19^2 - 13^2$$= 361 - 169$$= 192$
   3. Step 3: Square-root only at the end; it must be shorter than the hypotenuse$x = \sqrt{192}$$= 13.8564\ldots$$= 13.9$ cm (3 s.f.)Check: $13.9 < 19$, shorter than the hypotenuse.
5. Question 5Decimal lengths: name the side

   In triangle $STU$, $\angle STU = 90^\circ$, $SU = 10.4$ cm and $ST = 6.5$ cm. Find the length of $TU$, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: Name the hypotenuse: the side opposite the right angleThe hypotenuse is $SU$, opposite the right angle at $T$.
   2. Step 2: Use Pythagoras' theorem$SU^2 = ST^2 + TU^2$$TU^2 = 10.4^2 - 6.5^2$$= 108.16 - 42.25$$= 65.91$
   3. Step 3: Square-root only at the end; it must be shorter than the hypotenuse$TU = \sqrt{65.91}$$= 8.1184\ldots$$= 8.12$ cm (3 s.f.)Check: $8.12 < 10.4$, shorter than the hypotenuse.
6. Question 6Decimal lengths: name the side

   In triangle $LMN$, $\angle LMN = 90^\circ$, $LN = 13.6$ cm and $MN = 10.8$ cm. Find the length of $LM$, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: Name the hypotenuse: the side opposite the right angleThe hypotenuse is $LN$, opposite the right angle at $M$.
   2. Step 2: Use Pythagoras' theorem$LN^2 = LM^2 + MN^2$$LM^2 = 13.6^2 - 10.8^2$$= 184.96 - 116.64$$= 68.32$
   3. Step 3: Square-root only at the end; it must be shorter than the hypotenuse$LM = \sqrt{68.32}$$= 8.2655\ldots$$= 8.27$ cm (3 s.f.)Check: $8.27 < 13.6$, shorter than the hypotenuse.