# Given a right triangle triangle ABC with C=90^circ, if b=10, c=26, how do you find a?

Feb 24, 2017

$a = 24.0$

#### Explanation:

$\cos \angle A = \frac{a}{s} = \frac{10}{26} = 0.384615384$

arc cos angle A=67°22'48''

180°-(90°+67°22'48'')=22°37'12''=angleB

t/a=tan angle A=t/10 xx tan 67°22'48''

multiply L.H.S and R.H.S. by 10

t=C tan 67°22'48'' xx 10

$t = 2.399984066 \times 10 = 23.99984066$

$t = a = 24.0$

Check: using Pythagoras

$B {A}^{2} = C {A}^{2} + C {B}^{2}$

${26}^{2} = {10}^{2} + C {B}^{2}$

${10}^{2} + C {B}^{2} = {26}^{2}$

$C {B}^{2} = {26}^{2} - {10}^{2}$

$C B = \sqrt{{26}^{2} - {10}^{2}}$

$C B = \sqrt{676 - 100}$

$C B = \sqrt{576}$

$C B = a = 24.0$

Feb 26, 2017

Complete the ratio $5 \text{ : "12" : } 13$

to get $\text{ "10" : "color(blue)(24)" : } 26$

$a = 24$

#### Explanation:

There are right-angled triangles whose sides are rational numbers.

The sides are known as "Pythagorean Triples".

If you recognise that two given sides are in one of the triples, you can simply write down the length of the third side by simple multiplying.

Some of the common triples are:

$3 \text{ : "4" : } 5$
$5 \text{ : "12" : } 13$
$7 \text{ : "24" : } 25$
$8 \text{ : "15" : } 17$
$9 \text{ : "40" : } 41$
$11 \text{ : "60" : } 61$

Note that the following are all in the ratio: $\text{ "3" : "4" : } 5$

$\text{ "6" : "8" : "10" } \leftarrow \times 2$
$\text{ "9" : "12" : "15" } \leftarrow \times 3$
$\text{ "1.5" : "2" : "2.5" } \leftarrow \times 0.5$
$\text{ "7.5" : "10" : "12.5" } \leftarrow \times 2.5$
$\text{ "39" : "52" : "65" } \leftarrow \times 13$ and so on...

There are infinitely many triples which can be created.

In this case we have 2 sides as
$10 \text{ : } 26$ which are in the ratio $\text{ "5" : } 13$

The third side will therefore be 12 to complete the triple

$5 \text{ : "color(blue)(12)" : "13" } \leftarrow \times 2$

$10 \text{ : "color(blue)(24)" : } 26$