45 45 90 Triangle Rule

45-45-90 triangle

45-45-90 triangles are special right triangles with 1 $\mathrm{ninety}$ degree angle and 2 $45$ degree angles. All 45-45-90 triangles are considered special isosceles triangles. The 45-45-90 triangle has three unique properties that make it very special and dissimilar all the other triangles.

45-45-90 triangle ratio

At that place are ii ratios for 45-45-90 triangles:

• The ratio of the sides equals $1:1:\sqrt{2}$
• The ratio of the angles equals $\mathrm{one}:1:\mathrm{two}$

Properties of 45-45-xc triangles

To identify 45-45-90 special right triangle, check for these three identifying backdrop:

1. The polygon is an isosceles right triangle
2. The two side lengths are congruent, and their opposite angles are congruent
3. The hypotenuse (longest side) is the length of either leg times square root (sqrt) of 2, $\sqrt{\mathrm{two}}$

All 45-45-90 triangles are similar because they all have the same interior angles.

45-45-90 triangle theorem

To solve for the hypotenuse length of a 45-45-90 triangle, you can use the 45-45-ninety theorem, which says the length of the hypotenuse of a 45-45-xc triangle is the $\sqrt{2}$ times the length of a leg.

45-45-90 triangle formula

Y'all can also use the general form of the Pythagorean Theorem to observe the length of the hypotenuse of a 45-45-90 triangle.

Here is a 45-45-90 triangle. Let's use both methods to discover the unknown measure of a triangle where we only know the measure out of one leg is $59$ yards:

We can plug the known length of the leg into our 45-45-90 theorem formula:

$Hypot\mathrm{eastward}nuse=l\mathrm{due\; east}k\left(\sqrt{2}\right)$

$Hypotenus\mathrm{eastward}=59yds\left(\sqrt{2}\right)$

$\mathit{H}\mathit{y}\mathit{p}\mathit{o}\mathit{t}\mathit{due\; east}\mathit{n}\mathit{u}\mathit{s}\mathit{due\; east}\mathbf{}\mathbf{=}\mathbf{}\mathbf{83.4386}\mathbf{}\mathit{y}\mathit{d}\mathit{due\; south}$

Using the Pythagorean Theorem:

$a^{\mathrm{two}}+b^2=c^2$

${59}^{\mathrm{ii}}+{59}^2=c^2$

$\mathrm{three,481}yds+\mathrm{three,481}yd\mathrm{southward}=c^2$

$\mathrm{6,962}yd\mathrm{south}=c^2$

$\mathbf{83.4386}\mathbf{}\mathit{y}\mathit{d}\mathit{s}\mathbf{}\mathbf{=}\mathbf{}\mathit{c}$

Both methods produce the aforementioned consequence!

45-45-xc triangle rules

The main rule of 45-45-90 triangles is that it has one right bending and while the other 2 angles each measure $45°$.The lengths of the sides adjacent to the right triangle, the shorter sides have an equal length.

Another rule is that the two sides of the triangle or legs of the triangle that form the right angle are congruent in length.

Knowing these basic rules makes it easy to construct a 45-45-90 triangle.

Constructing a 45-45-90 Triangle

The easiest way to construct a 45-45-90 triangle is every bit follows:

1. Construct a square four equal sides to the desired length of the triangle's legs
2. Construct either diagonal of the foursquare

Striking the diagonal of the square creates two coinciding 45-45-ninety triangles. One-half of a square that has been cut by a diagonal is a 45-45-xc triangle.

The diagonal becomes the hypotenuse of a correct triangle.

You can also construct the triangle using a straightedge and drawing compass:

1. Construct a line segment more than twice every bit long as the desired length of your triangle'southward leg
2. Open the compass to span more than one-half the distance of the line segment
3. Use the compass to construct a perpendicular bisector of the line segment by scribing arcs from both endpoints above and below the line segment; this will produce 2 intersecting arcs above and 2 intersecting arcs below the line segment
4. Use the straightedge to draw the perpendicular bisector by connecting the intersecting arcs
5. Reset the compass with the point on the intersection of the two line segments and the span of the compass set to your desired length of the triangle'south leg
6. Strike two arcs, one on the line segment and 1 on the perpendicular bisector
7. Connect the intersections of the arcs and segments

This method takes more fourth dimension than the foursquare method but is elegant and does non require measuring.

How to solve a 45-45-ninety triangle

The length of the hypotenuse, which is the leg times $\sqrt{2}$, is primal to calculating the missing sides:

• If yous know the measure of the hypotenuse, divide the hypotenuse past $\sqrt{\mathrm{ii}}$ to find the length of either leg
• If y'all know the length of one leg, you know the length of the other leg (legs are congruent)
• If you know either leg's length, multiply the leg length times $\sqrt{2}$ to find the hypotenuse

45-45-90 triangle instance problems

Here we have a 45-45-90 trianlge with a hypotenuse of $3\sqrt{2}\mathrm{thousand}eters$ meters, and each leg is $3m\mathrm{eastward}ter\mathrm{south}$.

Recall, the hypotenuse is ever the measure of each leg times $\sqrt{\mathrm{ii}}$!

Take is a 45-45-90 triangle with sides measuring $9.5fe\mathrm{eastward}t$. What is the length of the hypotenuse?

You can reply either with $\mathrm{xiii.435}feet$, or with $\mathrm{ix.5}\sqrt{2}feet$.

What is the length of the hypotenuse in a 45-45-90 triangle with a leg of 10(√2) cm

The hypotenuse is $10\left(\sqrt{2}\right)\left(\sqrt{2}\right)$! Call up: what is $\left(\sqrt{2}\right)\left(\sqrt{2}\right)$? Information technology'south $2$, of course! And $10\times 2=20$. The hypotenuse of this triangle is $\mathrm{twenty}cm$.

Next Lesson:

30-60-ninety Triangle

45 45 90 Triangle Rule,

Source: https://tutors.com/math-tutors/geometry-help/45-45-90-triangle-rules-formula-theorem

Posted by: randallhatione.blogspot.com

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