Special Triangles

Special Triangles exercise

• Solve for the height and distance travelled on land of the plane.

  Hints

  The square root of a product is the product of the square roots of its factors:

  $\sqrt{a\times b}=\sqrt a \times \sqrt b$.

  The Pythagorean theorem says that the sum of the squares of both legs of a right triangle is the same like the square of the hypotenuse.

  An iscoceles triangle is one where two of its sides have the same length, while an equilateral triangle is one where all of its sides have the same length.

  Solution

  We are finding the base and height of an iscoceles right triangle: it has two angles of 45°, meaning that their opposite legs have the same length.

  So the hypotenuse is $c$ and both legs are $a$. Using the Pythagorean theorem, we get:

  $c^2=a^2+a^2=2a^2$

  Taking the square root of both sides, we get:

  $\sqrt{c^2}=\sqrt{2a^2}$

  which is equivalent to

  $c=\sqrt 2 ~ a$

  as the square root of a product is the product of the square roots of its factors.

  Plugging in the known value for $c=2450$, we get $2450=a~\sqrt 2$.

  And dividing by $\sqrt 2$ gives us $a=1,732.4\approx 1,732$ feet.

• Calculate the distance Grandpa Lindbergh's plane will fall.

  Hints

  The square root of a product is the product of the square roots of its factors.

  The Pythagorean theorem says that the sum of the squares of both legs of a right triangle is the same like the square of the hypotenuse.

  Solution

  We can think of an equilateral triangle as two 30-60-90 triangles, as pictured to the right. So if the sides of the equilateral triangle are $2a$, then the hypotenuse of the 30-60-90 triangles is $2a$ and their base is $a$. Let's call the height $b$.

  We can use the Pythagorean theorem to get $(2a)^2=a^2+b^2$. To square a product, we square each factor:

  $4a^2=a^2+b^2$.

  Subtracting $a^2$ from both sides leads to $3a^2=b^2$.

  Next we take the square root of both sides to get:

  $\sqrt {3a^2}=\sqrt {b^2}=b$.

  On the left-hand side we take the square root of each factor:

  $\sqrt 3~ a=b$.

  $b$ is the height of the plan before falling down, namely $b=1,732$.

  So we have to solve the following equation:

  $\begin{array}{rcl} \sqrt 3~a& = & ~1,732\\ \color{#669900}{\div\sqrt 3} & & \color{#669900}{\div\sqrt 3}\\ a & = & 1000 \end{array}$

  Multiplying $a$ by $2$, we get that the distance Grandpa Lindbergh's plane will fall is 2000 feet.

• Determine the length of the missing side.

  Hints

  Use the Pythagorean theorem: the sum of the squares of both legs of a right triangle is the same like the square of the hypotenuse.

  Determine the hypotenuse: the side opposite the right angle.

  For this example, the Pythagorean theorem says that $a^2=b^2+c^2$.

  Solution

  For each right triangle, use the Pythagorean theorem to find the length of the missing side:

  • Let's start with the green triangle: here we are missing the hypotenuse, $a$. The Pythagorean theorem gives us $a^2=3^2+4^2=9+16=25$, and taking the square root leads to $a=\sqrt{25}=5$.
  • Next the blue one with missing leg $c$. The hypotenuse has length $10$. Thus, $10^2=8^2+c^2$, which is equivalent to $100=64+c^2$. Subtracting $64$ from both sides and taking the square root leads to $c=\sqrt{36}=6$.
  • Now for the red triangle, with missing hypotenuse, $d$. We have that $d^2=4.5^2+6^2=56.25$. Taking the square root gives us $d=7.5$.
  • The last triangle is the yellow one, with a missing leg length $b$ and a hypotenuse length of $17$. This gives us $17^2=15^2+b^2$. Subtracting $225$ from both sides gives us $64=b^2$, and thus that $b=\sqrt{64}=8$.

• Find the distance Grandpa Lindbergh has to fly to get to a height of 1500 feet.

  Hints

  Decide if the missing side is the hypotenuse or not.

  Use the Pythagorean theorem: square each leg of a right triangle and add them together to get the square of the hypotenuse.

  Solution

  Look at the triangle to the right, with the missing hypotenuse $c$ and the legs of length 1500.

  Using the Pythagorean theorem, we get:

  $c^2=1500^2+1500^2=2\times 1500^2$

  So $c^2=4500000$.

  Taking the square root of both sides gives us the solution:

  $c=\sqrt{4500000}=2,121.32$

  Rounding the result gives us a distance of

  2,121 feet in which Grandpa Lindbergh needs to fly to get to a height of 1500 feet.

• State what a triangle with two 45° angles is called and what is special about it.

  Hints

  All angles of an equilateral triangle are equal.

  The sum of all the angles of any triangle is always 180°.

  The opposite legs of equal angles have equal lengths.

  Solution

  Any triangle with two 45° angles is an iscoceles right triangle.

  We can see this in the following way:

  • The sum of the three angles of any triangle is always 180°.
  • Thus, if two angles are 45°, the last angle must be $180^\circ - (45^\circ+45^\circ)=90^\circ$.
  • Also, the triangle legs opposite to the 45° angles have equal lengths.

• Calculate the length of the wooden slats needed to build the kite.

  Hints

  In an iscoceles right triangle, two legs have the same length and the third leg is the hypotenuse.

  You have to use the Pythagorean theorem twice.

  Keep in mind that in a 30-60-90 triangle the leg opposite to the right angle is double as long as the this one opposite the 30° angle.

  Solution

  First, we take a look at the isosceles right triangles on top. We know the length of the hypotenuse and thus we can use the Pythagorean theorem, where $a$ is the missing length:

  $12^2=2a^2$.

  Dividing by $2$ we get $a^2=72$, and so $a=\sqrt{72}=8.485...\approx 8.5$.

  We still need the length of the longer part of the middle axis, $b$, of the kite.

  We use the Pythagorean theorem, as well as the fact that the hypotenuse is twice as long as $a$: $2\times 8.5=17$.

  We then have that:

  $17^2=8.5^2+b^2$.

  Subtracting $72.25$, we get $b^2=216.75$ and thus $b=\sqrt{216.75}=14.72...\approx 14.7$.

  And so we know that the horizontal wooden slat much be $2\times 8.5=17$ inches, and the length of the vertical wooden slat must be $8.5+14.7=23.2$ inches.

  All together, the total amount of wood needed is $40,2$ inches.